This elliptic curve has smallest conductor amongst elliptic curves over $\Q$ of rank 3.
Minimal Weierstrass equation
\( y^2 + y = x^{3} - 7 x + 6 \)
Mordell-Weil group structure
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \( \left(-2, 3\right) \) | \( \left(-1, 3\right) \) | \( \left(0, 2\right) \) |
| \(\hat{h}(P)\) | ≈ | 1.36857250535 | 1.20508110419 | 0.990906333153 |
Integral points
\( \left(-3, 0\right) \), \( \left(-2, 3\right) \), \( \left(-1, 3\right) \), \( \left(0, 2\right) \), \( \left(1, 0\right) \), \( \left(2, 0\right) \), \( \left(3, 3\right) \), \( \left(4, 6\right) \), \( \left(8, 21\right) \), \( \left(11, 35\right) \), \( \left(14, 51\right) \), \( \left(21, 95\right) \), \( \left(37, 224\right) \), \( \left(52, 374\right) \), \( \left(93, 896\right) \), \( \left(342, 6324\right) \), \( \left(406, 8180\right) \), \( \left(816, 23309\right) \)
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 5077 \) | = | \(5077\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(5077 \) | = | \(5077 \) | ||
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{37933056}{5077} \) | = | \(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 5077^{-1}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
|
magma: Rank(E);
sage: E.rank()
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| Rank: | \(3\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(0.417143558758\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(4.15168798309\) | ||
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magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
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| Tamagawa product: | \( 1 \) = \( 1 \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(1\) | ||
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magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (rounded) | ||
Modular invariants
Modular form 5077.2.a.a
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 1984 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L^{(3)}(E,1)/3! \) ≈ \( 1.73184990012 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(5077\) | \(1\) | \( I_{1} \) | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) .
$p$-adic data
$p$-adic regulators
Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 5077 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ss | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ss | ss | ordinary | ordinary | nonsplit |
| $\lambda$-invariant(s) | 4,3 | 3,3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3,3 | 3,3 | 3 | 3 | ? |
| $\mu$-invariant(s) | 0,0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0,0 | 0 | 0 | ? |
An entry ? indicates that the invariants have not yet been computed.
Isogenies
This curve has no rational isogenies. Its isogeny class 5077a consists of this curve only.
Growth of torsion in number fields
The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 3 | 3.3.20308.1 | \(\Z/2\Z\) | Not in database |
| 6 | 6.6.2093830264528.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.
Additional information
Historical Information about the Gauss elliptic curve
In 1985, Buhler, Gross and Zagier used the celebrated Gross-Zagier Theorem on heights of Heegner points (see Gross, Benedict H.; Zagier, Don B. (1986), "Heegner points and derivatives of L-series", Inventiones Mathematicae 84 (2): 225–320, [10.1007/BF01388809]) to prove that the L-function of this curve has a zero of order 3 at its critical point $s=1$, thus establishing the first part of the Birch and Swinnerton-Dyer conjecture for this curve (see Math. Comp. 44 (1985), 473-481: [10.1090/S0025-5718-1985-0777279-X]). This was the first time that BSD had been established for any elliptic curve of rank $3$. To this day, it is not possible, even in principle, to establish BSD for any curve of rank $4$ or greater since there is no known method for rigourously establishing the value of the analytic rank when it is greater than $3$.
Via Goldfeld's method, which required the use of an L-function of analytic rank at least $3$, this elliptic curve also found an application in the context of obtaining explicit lower bounds for the class numbers of imaginary quadratic fields. This solved Gauss's Class Number Problem first posed by Gauss in 1801 is his book Disquisitiones Arithmeticae (Section V, Articles 303 and 304).