Minimal Weierstrass equation
\( y^2 + x y = x^{3} + 8938334 x + 2268820292 \)
Mordell-Weil group structure
Torsion generators
\( \left(-28, 44942\right) \)
Integral points
\( \left(-28, 44942\right) \), \( \left(1844, 157262\right) \), \( \left(9956, 1033358\right) \)
Invariants
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magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 101478 \) | = | \(2 \cdot 3 \cdot 13 \cdot 1301\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(-47925805879636550221824 \) | = | \(-1 \cdot 2^{28} \cdot 3^{7} \cdot 13^{7} \cdot 1301 \) | ||
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{78975693098270145722349791}{47925805879636550221824} \) | = | \(2^{-28} \cdot 3^{-7} \cdot 7^{6} \cdot 13^{-7} \cdot 31^{3} \cdot 41^{3} \cdot 83^{6} \cdot 1301^{-1}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
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magma: Rank(E);
sage: E.rank()
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| Rank: | \(0\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(1\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(0.069533620702\) | ||
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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: | \( 1372 \) = \( ( 2^{2} \cdot 7 )\cdot7\cdot7\cdot1 \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(7\) | ||
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magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(4\) (exact) | ||
Modular invariants
Modular form 101478.2.a.k
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 9985024 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 7.78776551862 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(28\) | \( I_{28} \) | Split multiplicative | -1 | 1 | 28 | 28 |
| \(3\) | \(7\) | \( I_{7} \) | Split multiplicative | -1 | 1 | 7 | 7 |
| \(13\) | \(7\) | \( I_{7} \) | Split multiplicative | -1 | 1 | 7 | 7 |
| \(1301\) | \(1\) | \( I_{1} \) | 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 \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(7\) | B.1.1 |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 13 | 1301 |
|---|---|---|---|---|---|---|
| Reduction type | split | split | ordinary | ordinary | split | nonsplit |
| $\lambda$-invariant(s) | 3 | 1 | 0 | 4 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class 101478k
consists of 2 curves linked by isogenies of
degree 7.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 3 | 3.1.50739.3 | \(\Z/14\Z\) | Not in database |
| 6 | 6.0.130624821733419.4 | \(\Z/2\Z \times \Z/14\Z\) | 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.