Minimal Weierstrass equation
\( y^2 + x y = x^{3} - x^{2} - 78417 x - 10666634 \)
Mordell-Weil group structure
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \( \left(434, -6292\right) \) | \( \left(\frac{30986}{49}, -\frac{4865176}{343}\right) \) |
| \(\hat{h}(P)\) | ≈ | 1.64543563 | 4.5393361152 |
Torsion generators
\( \left(\frac{1331}{4}, -\frac{1331}{8}\right) \)
Integral points
\( \left(434, 5858\right) \), \( \left(1178, 38534\right) \), \( \left(2834, 148658\right) \), \( \left(99578, 31372784\right) \)
Invariants
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magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 150075 \) | = | \(3^{2} \cdot 5^{2} \cdot 23 \cdot 29\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(-18471269925703125 \) | = | \(-1 \cdot 3^{12} \cdot 5^{7} \cdot 23^{2} \cdot 29^{2} \) | ||
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( -\frac{4681768588489}{1621620405} \) | = | \(-1 \cdot 3^{-6} \cdot 5^{-1} \cdot 23^{-2} \cdot 29^{-2} \cdot 16729^{3}\) | ||
| 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: | \(2\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(6.44864473252\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(0.140199755984\) | ||
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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: | \( 64 \) = \( 2^{2}\cdot2^{2}\cdot2\cdot2 \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(2\) | ||
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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 150075.2.a.bj
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 663552 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L^{(2)}(E,1)/2! \) ≈ \( 14.4655746868 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(4\) | \( I_6^{*} \) | Additive | -1 | 2 | 12 | 6 |
| \(5\) | \(4\) | \( I_1^{*} \) | Additive | 1 | 2 | 7 | 1 |
| \(23\) | \(2\) | \( I_{2} \) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(29\) | \(2\) | \( I_{2} \) | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 X6.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.
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 |
$p$-adic data
$p$-adic regulators
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class 150075bj
consists of 2 curves linked by isogenies of
degree 2.
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 | \(\Q(\sqrt{-5}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| 4 | \( x^{4} + x^{2} - 500 \) | \(\Z/4\Z\) | 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.