Minimal Weierstrass equation
\(y^2=x^3+x^2-10145x+679615\)
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
\(P\) | = | \(\left(\frac{1147}{9}, \frac{32768}{27}\right)\) ![]() |
\(\hat{h}(P)\) | ≈ | $1.6023759029409812598275046359$ |
Integral points
\(\)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 15680 \) | = | \(2^{6} \cdot 5 \cdot 7^{2}\) |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | \(-134690174402560 \) | = | \(-1 \cdot 2^{39} \cdot 5 \cdot 7^{2} \) |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( -\frac{8990558521}{10485760} \) | = | \(-1 \cdot 2^{-21} \cdot 5^{-1} \cdot 7 \cdot 1087^{3}\) |
Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | \(1\) | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | \(1.6023759029409812598275046359\) | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | \(0.52876102563007103336284358876\) | ||
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: | \( 4 \) = \( 2^{2}\cdot1\cdot1 \) | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | \(1\) | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | \(1\) (exact) |
Modular invariants
Modular form 15680.2.a.t

For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 48384 | ||
\( \Gamma_0(N) \)-optimal: | no | ||
Manin constant: | 1 |
Special L-value
\( L'(E,1) \) ≈ \( 3.3890957035359376258157051922158169935 \)
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(2\) | \(4\) | \(I_{29}^{*}\) | Additive | 1 | 6 | 39 | 21 |
\(5\) | \(1\) | \(I_{1}\) | Split multiplicative | -1 | 1 | 1 | 1 |
\(7\) | \(1\) | \(II\) | Additive | -1 | 2 | 2 | 0 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
prime | Image of Galois representation |
---|---|
\(3\) | B |
$p$-adic data
$p$-adic regulators
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | ordinary | split | add | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
$\lambda$-invariant(s) | - | 5 | 2 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | - | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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 \(d=\)
3.
Its isogeny class 15680.t
consists of 2 curves linked by isogenies of
degree 3.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{42}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.1960.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.153664000.2 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
$6$ | 6.0.3920736960000.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.2.23233996800.1 | \(\Z/6\Z\) | Not in database |
$12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | Deg 12 | \(\Z/3\Z \times \Z/3\Z\) | Not in database |
$12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
$18$ | 18.6.121477087050738706656707080303411200000000.2 | \(\Z/9\Z\) | Not in database |
$18$ | 18.0.96432428246090143347612057600000000000000.1 | \(\Z/6\Z\) | 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.