Minimal Weierstrass equation
\(y^2+xy+y=x^3-24917x+26570126\)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(-198, 4972\right) \) |
| \(\hat{h}(P)\) | ≈ | $0.49201847483060297332053064833$ |
Torsion generators
\( \left(-\frac{1305}{4}, \frac{1301}{8}\right) \)
Integral points
\( \left(-198, 4972\right) \), \( \left(-198, -4775\right) \), \( \left(36, 5053\right) \), \( \left(36, -5090\right) \), \( \left(486, 11128\right) \), \( \left(486, -11615\right) \), \( \left(1968, 86197\right) \), \( \left(1968, -88166\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 348726 \) | = | \(2 \cdot 3 \cdot 7 \cdot 19^{2} \cdot 23\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-304037387526334518 \) | = | \(-1 \cdot 2 \cdot 3^{8} \cdot 7^{2} \cdot 19^{7} \cdot 23^{2} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{36363385297}{6462571878} \) | = | \(-1 \cdot 2^{-1} \cdot 3^{-8} \cdot 7^{-2} \cdot 19^{-1} \cdot 23^{-2} \cdot 3313^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(1\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(0.49201847483060297332053064833\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.25056101600258031010782778095\) | ||
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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: | \( 128 \) = \( 1\cdot2^{3}\cdot2\cdot2^{2}\cdot2 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(2\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 348726.2.a.p
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 4792320 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 3.9449807662590678122130069243987250458 \)
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(3\) | \(8\) | \(I_{8}\) | Split multiplicative | -1 | 1 | 8 | 8 |
| \(7\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(19\) | \(4\) | \(I_1^{*}\) | Additive | -1 | 2 | 7 | 1 |
| \(23\) | \(2\) | \(I_{2}\) | 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 348726.p
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-38}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $4$ | 4.2.15759968.3 | \(\Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | 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.