Minimal Weierstrass equation
\(y^2+xy=x^3+9096x+224832\)
Mordell-Weil group structure
trivial
Integral points
\(\)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 966 \) | = | \(2 \cdot 3 \cdot 7 \cdot 23\) |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | \(-69854999176704 \) | = | \(-1 \cdot 2^{9} \cdot 3 \cdot 7^{11} \cdot 23 \) |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{83228502970940543}{69854999176704} \) | = | \(2^{-9} \cdot 3^{-1} \cdot 7^{-11} \cdot 23^{-1} \cdot 436607^{3}\) |
Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | \(1.3441305674074603534852663448\dots\) | ||
Stable Faltings height: | \(1.3441305674074603534852663448\dots\) |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | \(0\) | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | \(1\) | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | \(0.39928077868365073906929752176\dots\) | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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Tamagawa product: | \( 9 \) = \( 3^{2}\cdot1\cdot1\cdot1 \) | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | \(1\) | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | \(1\) (exact) |
Modular invariants

For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 3960 | ||
\( \Gamma_0(N) \)-optimal: | yes | ||
Manin constant: | 1 |
Special L-value
\( L(E,1) \) ≈ \( 3.5935270081528566516236776958316971085 \)
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(2\) | \(9\) | \(I_{9}\) | Split multiplicative | -1 | 1 | 9 | 9 |
\(3\) | \(1\) | \(I_{1}\) | Split multiplicative | -1 | 1 | 1 | 1 |
\(7\) | \(1\) | \(I_{11}\) | Non-split multiplicative | 1 | 1 | 11 | 11 |
\(23\) | \(1\) | \(I_{1}\) | 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
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | ordinary | nonsplit | ordinary | ordinary | ordinary | ss | split | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
$\lambda$-invariant(s) | 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0,0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Isogenies
This curve has no rational isogenies. Its isogeny class 966.k consists of this curve only.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 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 |
---|---|---|---|
$3$ | 3.1.3864.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.57691436544.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
$8$ | 8.2.119024757698427.3 | \(\Z/3\Z\) | Not in database |
$12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.