Minimal Weierstrass equation
\( y^2 = x^{3} - x^{2} - 964481 x + 364797825 \)
Mordell-Weil group structure
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(673, 4480\right) \) |
| \(\hat{h}(P)\) | ≈ | 2.25903904516 |
Torsion generators
\( \left(575, 0\right) \)
Integral points
\( \left(575, 0\right) \), \((673,\pm 4480)\)
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 6720 \) | = | \(2^{6} \cdot 3 \cdot 5 \cdot 7\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(31596430963507200 \) | = | \(2^{22} \cdot 3^{16} \cdot 5^{2} \cdot 7 \) | ||
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{378499465220294881}{120530818800} \) | = | \(2^{-4} \cdot 3^{-16} \cdot 5^{-2} \cdot 7^{-1} \cdot 723361^{3}\) | ||
| 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: | \(1\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(2.25903904516\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(0.362722094214\) | ||
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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: | \( 16 \) = \( 2^{2}\cdot2\cdot2\cdot1 \) | ||
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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: | \(1\) (exact) | ||
Modular invariants
Modular form 6720.2.a.j
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 98304 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 3.27761349349 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(4\) | \( I_12^{*} \) | Additive | 1 | 6 | 22 | 4 |
| \(3\) | \(2\) | \( I_{16} \) | Non-split multiplicative | 1 | 1 | 16 | 16 |
| \(5\) | \(2\) | \( I_{2} \) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(7\) | \(1\) | \( I_{1} \) | Non-split multiplicative | 1 | 1 | 1 | 1 |
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 X235k.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 9 \end{array}\right)$ and has index 96.
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.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | nonsplit | nonsplit | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ss | ordinary | ordinary | ordinary | ss |
| $\lambda$-invariant(s) | - | 1 | 3 | 1 | 1 | 1 | 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 | 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 \(d=\)
2, 4, 8 and 16.
Its isogeny class 6720.j
consists of 8 curves linked by isogenies of
degrees dividing 16.
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 | \(\Q(\sqrt{2}) \) | \(\Z/8\Z\) | Not in database |
| \(\Q(\sqrt{7}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database | |
| \(\Q(\sqrt{14}) \) | \(\Z/4\Z\) | Not in database | |
| 4 | \(\Q(\sqrt{2}, \sqrt{35})\) | \(\Z/16\Z\) | Not in database |
| \(\Q(\sqrt{2}, \sqrt{7})\) | \(\Z/2\Z \times \Z/8\Z\) | Not in database | |
| \(\Q(\sqrt{2}, \sqrt{5})\) | \(\Z/16\Z\) | 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.