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Simplest cubic fields with small class number

This paper uses computational methods to enumerate and explicitly list integers $m$ within specific ranges that yield simplest cubic fields with small class numbers, categorizing the results based on the index of the ring of integers and the conductor.

Original authors: Akinari Hoshi, Hiroaki Iida

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Akinari Hoshi, Hiroaki Iida

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect designing a special kind of building called a "Simplest Cubic Field."

In the world of mathematics, these buildings are constructed using a specific blueprint (a formula) involving a number we'll call $m$ . The blueprint looks like this:
$x^3 - mx^2 - (m+3)x - 1 = 0$

Every time you pick a different integer for $m$ (like -1, 0, 1, 2, 3...), you get a unique building. These buildings are fascinating because they are "cyclic," meaning they have a perfect, repeating symmetry, much like a three-bladed fan or a triangle.

The Two Big Questions

The authors of this paper, Hoshi and Iida, are like detectives trying to answer two main questions about these buildings:

How "messy" is the interior?
In math, the "messiness" of a building is called its Class Number ( $h_m$ ).
- If the class number is 1, the building is perfectly organized. Every room fits together perfectly; there are no loose bricks or confusing hallways. It's a "Principal Ideal Domain."
- If the class number is 3, 4, 7, 100, the building has some structural quirks. It's still standing, but it has "holes" in its organization that require extra keys to unlock certain doors.
- The higher the number, the more chaotic the interior.
Is the foundation solid?
The foundation is the relationship between the blueprint ( $m$ ) and the actual materials used to build the walls (the "Ring of Integers").
- Sometimes the blueprint matches the materials perfectly (Index = 1).
- Sometimes the materials are slightly off, requiring a "scaffolding" factor of 3 or 27 to make sense of the structure.

The Great Census

The authors went on a massive digital expedition using a powerful calculator called PARI/GP. They didn't just look at a few buildings; they checked millions of them.

What they found:

The "Perfect" Buildings (Class Number 1):
They found exactly 26 special values of $m$ (up to 4 million) where the building is perfectly organized ( $h_m = 1$ ).
- Analogy: Imagine looking for a needle in a haystack. They found 26 needles in a haystack the size of a city.
- Some famous needles are $m = -1, 0, 1, 2, 3, 4, 7, 8, 10, 12, \dots$
The "Slightly Messy" Buildings:
They also counted how many buildings had small amounts of messiness ( $h_m = 3, 4, 7, 9, 12, 13$ ).
- There were 31 buildings with messiness level 3.
- There were 10 with level 4.
- And so on.
- In total, they found 137 buildings with very low messiness (less than 16) within their search range.

The "Scaffolding" Factor

The paper also looked at how the blueprint relates to the materials.

Case 1 (Index 1): The blueprint fits perfectly. They found 581 such buildings with low messiness.
Case 2 (Index 3): The blueprint needs a little extra help (a factor of 3). They found 80 such buildings.
Case 3 (Index 27): The blueprint needs a lot of scaffolding (a factor of 27). They found 142 such buildings.

Why Does This Matter?

You might ask, "Who cares about counting messy math buildings?"

The "Finite" Mystery: The authors proved that for any specific level of messiness (say, exactly 1000), there are only a finite number of these buildings. You can't keep finding new ones forever; eventually, the buildings get so huge and complex that their messiness explodes.
The "GRH" Safety Net: To do their calculations, they used a famous mathematical guess called the Generalized Riemann Hypothesis (GRH). It's like using a trusted map that might be slightly wrong, but is usually right.
- The Twist: For the specific buildings they found with very low messiness, they used a "certification" tool to prove the map was actually correct. They didn't need the guess; they had the proof.
The Missing Pieces: They corrected a small error in a previous study (by Lettl in 1986), finding a building that was accidentally left off the list (specifically, the one where $m=12$ ).

The Big Picture Analogy

Imagine you are cataloging every type of snowflake that can ever exist.

Most snowflakes are complex and unique.
But you are only interested in the "simplest" ones that have a perfect 3-fold symmetry.
You want to know: "How many of these simple snowflakes have a perfect, unbroken crystal structure?"
Hoshi and Iida went out, scanned the entire universe of these snowflakes (up to a certain size), and made a definitive list. They said, "Here are the 26 perfect ones. Here are the 31 that are almost perfect. And here is the proof that we haven't missed any others in this range."

Summary

This paper is a comprehensive census of a specific family of mathematical structures. The authors used modern computers to:

Count exactly how many "simplest cubic fields" have small class numbers.
Provide a complete list of the specific numbers ( $m$ ) that create them.
Prove that this list is complete for the range they checked, giving mathematicians a solid foundation for future discoveries.

It's a mix of heavy number theory, computer power, and the satisfaction of knowing that, for now, we have found all the "perfect" and "nearly perfect" members of this specific mathematical family.

1. Problem Statement

The paper investigates the class numbers ( $h_m$ ) of the simplest cubic fields $L_m = \mathbb{Q}(\alpha)$ , where $\alpha$ is a root of the polynomial:
$f_m(X) = X^3 - mX^2 - (m+3)X - 1$
for integers $m \in \mathbb{Z}$ . These fields are cyclic cubic extensions of $\mathbb{Q}$ with Galois group $C_3$ .

The authors aim to:

Determine the exact number of integers $m \ge -1$ such that the class number $h_m \le 1000$ , categorized by the index $[\mathcal{O}_{L_m} : \mathbb{Z}[\alpha]]$ (which relates to whether the polynomial is monogenic).
Identify all integers $m$ in the range $-1 \le m \le 4 \cdot 10^6$ where the class number is small ( $h_m < 16$ ).
Provide explicit lists of these integers and the structure of their ideal class groups.

2. Methodology

The authors employ a combination of theoretical number theory and extensive computational verification using PARI/GP.

Theoretical Framework

Field Properties: The discriminant of $f_m(X)$ is $(m^2 + 3m + 9)^2$ . The conductor $f_m$ of the field $L_m$ is derived from $m^2 + 3m + 9$ based on the prime factorization and congruence conditions of $m$ modulo 3, 9, and 27.
Index Calculation: The index $[\mathcal{O}_{L_m} : \mathbb{Z}[\alpha]]$ $[O_{L_{m}} : Z [α]]$ determines the relationship between the discriminant of the polynomial and the field discriminant. The paper focuses on three specific cases:
- Index = 1 (Monogenic): $m^2 + 3m + 9 = f_m$
- Index = 3: $m^2 + 3m + 9 = 3f_m$
- Index = 27: $m^2 + 3m + 9 = 27f_m$
Bounds: The authors utilize Lettl's lower bound for the class number (derived from the class number formula and regulator bounds) to establish upper limits for $m$ . Specifically, for a fixed index and a fixed class number $h$ , there are only finitely many $m$ . This allows the search space to be truncated (e.g., $m \le 5762$ for index 1 and $h \le 1000$ ).
Thue Equations: The paper references prior work (Hoshi, Okazaki) on field coincidences ( $L_m = L_n$ ) and Thue equations to ensure the uniqueness of the fields being analyzed.

Computational Approach

Software: All computations were performed using PARI/GP.
Algorithms:
- bnfinit: Computes the number field data, including the class number and class group structure.
- bnfcertify: Used to verify the computed class numbers unconditionally (without assuming the Generalized Riemann Hypothesis - GRH) for the specific ranges where the bounds allow.
- For the larger range ( $m \le 4 \cdot 10^6$ ), computations were initially done under GRH and then certified for the subset where $h_m < 16$ .
Search Strategy: The authors iterated through integers $m$ satisfying specific congruence conditions (modulo 3, 9, 27) and primality/square-free conditions for the conductor, filtering for those with small class numbers.

3. Key Contributions and Results

A. Classification for $h_m \le 1000$ (Theorem 1.12)

The authors provide a complete enumeration of integers $m \ge -1$ yielding class numbers up to 1000, split by the index $[\mathcal{O}_{L_m} : \mathbb{Z}[\alpha]]$ :

Index = 1 (Monogenic case, $m^2+3m+9 = f_m$ ):
- Total count: 581 integers.
- Breakdown:
  - $m=0$ ( $h=1$ ).
  - $m \not\equiv 0 \pmod 3$ with prime conductor: 149 integers (7 have $h=1$ ).
  - $m \not\equiv 0 \pmod 3$ with composite conductor: 308 integers (5 have $h=3$ ).
  - $m \equiv 0, 6 \pmod 9$ with conductor $9p$ : 56 integers (4 have $h=3$ ).
  - $m \equiv 0, 6 \pmod 9$ with conductor $9s$ : 67 integers.
Index = 3 ( $m^2+3m+9 = 3f_m$ ):
- Total count: 80 integers.
- Breakdown:
  - $m=3$ ( $h=1$ ).
  - $m \equiv 3, 21 \pmod{27}$ with conductor $9p$ : 34 integers (2 have $h=3$ ).
  - $m \equiv 3, 21 \pmod{27}$ with conductor $9s$ : 45 integers.
Index = 27 ( $m^2+3m+9 = 27f_m$ ):
- Total count: 142 integers.
- Breakdown:
  - $m \equiv 12 \pmod{27}$ with prime conductor: 45 integers (3 have $h=1$ ).
  - $m \equiv 12 \pmod{27}$ with composite conductor: 97 integers (2 have $h=3$ ).

B. Classification for $h_m < 16$ up to $4 \cdot 10^6$ (Theorem 1.13)

The authors extended the search to $m \le 4 \cdot 10^6$ to find all instances of very small class numbers.

Total Count: There are exactly 137 integers in this range with $h_m < 16$ .
Distribution:
- $h_m = 1$ : 26 integers (including $m = -1, 0, 1, 2, 3, 4, 5, 7, 8, 10, 12, 39, 54, 66, 93, \dots$ ).
- $h_m = 3$ : 31 integers.
- $h_m = 4$ : 10 integers.
- $h_m = 7$ : 10 integers.
- $h_m = 9$ : 36 integers.
- $h_m = 12$ : 21 integers.
- $h_m = 13$ : 3 integers.
Significance of the Bound: The paper notes that under the Generalized Riemann Hypothesis (GRH), these 137 integers are the only solutions for $h_m < 16$ in the entire range $-1 \le m \le 4 \cdot 10^6$ .

C. Explicit Data

The paper includes extensive tables (Tables 1–7) listing the specific values of $m$ , their congruence classes, the value of $m^2+3m+9$ , its prime factorization, and the structure of the ideal class group (e.g., $C_2 \times C_2$ , $C_3$ , etc.) for every case found.

4. Significance

Completeness: This work provides the most comprehensive dataset to date for the class numbers of simplest cubic fields, moving beyond previous partial results (e.g., Lettl, Byeon) which were limited to specific sub-cases or smaller ranges.
Monogenicity: By separating results based on the index $[\mathcal{O}_{L_m} : \mathbb{Z}[\alpha]]$ , the paper clarifies the distribution of class numbers specifically for monogenic fields (Index 1), a topic of significant interest in algebraic number theory.
Verification of Conjectures: The results confirm and extend earlier theoretical bounds. The explicit listing of all $m$ with $h_m < 16$ up to $4 \cdot 10^6$ serves as a crucial reference for testing conjectures regarding the distribution of class numbers in cyclic cubic fields.
Correction of Literature: The authors explicitly correct a missing case ( $m=12$ with $h=1$ ) in a previous result by Lettl (1986), demonstrating the necessity of rigorous computational verification.
Methodological Benchmark: The paper serves as a benchmark for computational number theory, demonstrating how to combine analytic bounds with modern computer algebra systems to solve Diophantine problems in number fields.

In summary, Hoshi and Iida have effectively mapped the "landscape" of small class numbers for simplest cubic fields, providing a definitive list of parameters $m$ that generate these fields, thereby advancing the understanding of the arithmetic properties of this specific family of cyclic cubic extensions.