Minimal Weierstrass equation
\(y^2+xy=x^3+x^2-16014352x-24017530676\)
Mordell-Weil group structure
\(\Z/{2}\Z \times \Z/{2}\Z\)
Torsion generators
\( \left(-2612, 1306\right) \), \( \left(-1996, 998\right) \)
Integral points
\( \left(-2612, 1306\right) \), \( \left(-1996, 998\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 177870 \) | = | \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(13792885084738553062500 \) | = | \(2^{2} \cdot 3^{2} \cdot 5^{6} \cdot 7^{12} \cdot 11^{6} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{2179252305146449}{66177562500} \) | = | \(2^{-2} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-6} \cdot 13^{3} \cdot 9973^{3}\) |
| 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: | \(0\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.075609988161904240662844212173\) | ||
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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: | \( 384 \) = \( 2\cdot2\cdot( 2 \cdot 3 )\cdot2^{2}\cdot2^{2} \) | ||
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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 177870.2.a.bs
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: | 19906560 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 1.8146397158857017759082610921551095711 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(5\) | \(6\) | \(I_{6}\) | Split multiplicative | -1 | 1 | 6 | 6 |
| \(7\) | \(4\) | \(I_6^{*}\) | Additive | -1 | 2 | 12 | 6 |
| \(11\) | \(4\) | \(I_0^{*}\) | 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 X8.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $$ 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 |
|---|---|
| \(2\) | Cs |
| \(3\) | B |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class 177870hp
consists of 4 curves linked by isogenies of
degrees dividing 12.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-231}) \) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{11}, \sqrt{105})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-11}, \sqrt{-14})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{30}, \sqrt{-154})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $6$ | 6.2.431325073872.4 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $8$ | 8.0.455583411360000.48 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $8$ | 8.0.11662935330816.5 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $8$ | 8.0.7289334581760000.435 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/6\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $18$ | 18.0.432159117806261814937856271170958531937500000000.1 | \(\Z/2\Z \times \Z/18\Z\) | 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.