Minimal Weierstrass equation
\(y^2=x^3-x^2-16x-222284\)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
\(P\) | = | \( \left(68, 294\right) \) |
\(\hat{h}(P)\) | ≈ | $1.5890007130183937822112027149$ |
Torsion generators
\( \left(61, 0\right) \)
Integral points
\( \left(61, 0\right) \), \((68,\pm 294)\), \((1636,\pm 66150)\)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 23520 \) | = | \(2^{5} \cdot 3 \cdot 5 \cdot 7^{2}\) |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | \(-21346234560000 \) | = | \(-1 \cdot 2^{9} \cdot 3^{4} \cdot 5^{4} \cdot 7^{7} \) |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( -\frac{8}{354375} \) | = | \(-1 \cdot 2^{3} \cdot 3^{-4} \cdot 5^{-4} \cdot 7^{-1}\) |
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: | \(1.5890007130183937822112027149\) | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | \(0.31189881823201773973332386518\) | ||
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: | \( 32 \) = \( 2\cdot2\cdot2\cdot2^{2} \) | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | \(2\) | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | \(1\) (exact) |
Modular invariants
Modular form 23520.2.a.l
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 73728 | ||
\( \Gamma_0(N) \)-optimal: | no | ||
Manin constant: | 1 |
Special L-value
\( L'(E,1) \) ≈ \( 3.9648595564821646943951303083721289154 \)
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(2\) | \(2\) | \(I_0^{*}\) | Additive | -1 | 5 | 9 | 0 |
\(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | 1 | 1 | 4 | 4 |
\(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | 1 | 1 | 4 | 4 |
\(7\) | \(4\) | \(I_1^{*}\) | Additive | -1 | 2 | 7 | 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 X13e.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.
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 | add | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
$\lambda$-invariant(s) | - | 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 |
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 and 4.
Its isogeny class 23520.l
consists of 3 curves linked by isogenies of
degrees dividing 4.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-14}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-7})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
$8$ | 8.0.1973822685184.10 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
$8$ | 8.4.24981193359360000.46 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.31863767040000.192 | \(\Z/8\Z\) | Not in database |
$8$ | Deg 8 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/12\Z\) | 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.