Minimal Weierstrass equation
\( y^2 + x y + y = x^{3} + x^{2} - 1011018 x - 392829865 \)
Mordell-Weil group structure
Torsion generators
\( \left(1161, -581\right) \)
Integral points
\( \left(1161, -581\right) \)
Invariants
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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Conductor: | \( 11858 \) | = | \(2 \cdot 7^{2} \cdot 11^{2}\) | ||
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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Discriminant: | \(-382456734842355712 \) | = | \(-1 \cdot 2^{18} \cdot 7^{7} \cdot 11^{6} \) | ||
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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j-invariant: | \( -\frac{548347731625}{1835008} \) | = | \(-1 \cdot 2^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 1637^{3}\) | ||
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: | \(0\) | ||
magma: Regulator(E);
sage: E.regulator()
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Regulator: | \(1\) | ||
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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Real period: | \(0.075264957159\) | ||
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: | \( 288 \) = \( ( 2 \cdot 3^{2} )\cdot2^{2}\cdot2^{2} \) | ||
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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Torsion order: | \(2\) | ||
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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Analytic order of Ш: | \(1\) (exact) |
Modular invariants
Modular form 11858.2.a.bm
magma: ModularDegree(E);
sage: E.modular_degree()
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Modular degree: | 207360 | ||
\( \Gamma_0(N) \)-optimal: | no | ||
Manin constant: | 1 |
Special L-value
\( L(E,1) \) ≈ \( 5.41907691545 \)
Local data
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(2\) | \(18\) | \( I_{18} \) | Split multiplicative | -1 | 1 | 18 | 18 |
\(7\) | \(4\) | \( I_1^{*} \) | Additive | -1 | 2 | 7 | 1 |
\(11\) | \(4\) | \( I_0^{*} \) | Additive | -1 | 2 | 6 | 0 |
Galois representations
The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X16.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}\right)$ and has index 6.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
prime | Image of Galois representation |
---|---|
\(2\) | B |
\(3\) | B |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
$p$ | 2 | 3 | 7 | 11 |
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Reduction type | split | ordinary | add | add |
$\lambda$-invariant(s) | 4 | 0 | - | - |
$\mu$-invariant(s) | 0 | 2 | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 6, 9 and 18.
Its isogeny class 11858bk
consists of 6 curves linked by isogenies of
degrees dividing 18.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
---|---|---|---|
2 | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
\(\Q(\sqrt{-231}) \) | \(\Z/6\Z\) | Not in database | |
4 | \(\Q(\sqrt{-7}, \sqrt{33})\) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
4.2.54208.3 | \(\Z/4\Z\) | Not in database | |
6 | 6.0.440311012911.5 | \(\Z/18\Z\) | Not in database |
6.2.16307815293.1 | \(\Z/6\Z\) | 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.