Minimal Weierstrass equation
\(y^2=x^3+x^2-1694040x-688445100\)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z \times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(-642, 11616\right) \) |
| \(\hat{h}(P)\) | ≈ | $2.3635422827840807581521731305$ |
Torsion generators
\( \left(-466, 0\right) \), \( \left(1470, 0\right) \)
Integral points
\( \left(-1005, 0\right) \), \((-642,\pm 11616)\), \( \left(-466, 0\right) \), \( \left(1470, 0\right) \), \((2670,\pm 117600)\), \((5463,\pm 391314)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 203280 \) | = | \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(106723147611548160000 \) | = | \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{74093292126001}{14707625625} \) | = | \(3^{-4} \cdot 5^{-4} \cdot 7^{-4} \cdot 11^{-2} \cdot 97^{3} \cdot 433^{3}\) |
| 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: | \(2.3635422827840807581521731305\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.13416848537048463407791788461\) | ||
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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: | \( 512 \) = \( 2^{2}\cdot2^{2}\cdot2^{2}\cdot2\cdot2^{2} \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(4\) | ||
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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 203280.2.a.gb
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: | 7864320 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 10.147612422087609446524011006619316278 \)
Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(4\) | \(I_4^{*}\) | Additive | -1 | 4 | 12 | 0 |
| \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | -1 | 1 | 4 | 4 |
| \(5\) | \(4\) | \(I_{4}\) | Split multiplicative | -1 | 1 | 4 | 4 |
| \(7\) | \(2\) | \(I_{4}\) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(11\) | \(4\) | \(I_2^{*}\) | Additive | -1 | 2 | 8 | 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 X25i.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 0 & 1 \end{array}\right)$ and has index 24.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | Cs |
$p$-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 and 4.
Its isogeny class 203280x
consists of 3 curves linked by isogenies of
degrees dividing 8.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(i, \sqrt{11})\) | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.0.7289334581760000.30 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.4.5877614223360000.10 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | 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.