Minimal Weierstrass equation
\(y^2=x^3+x^2-1664x-9804\)
Mordell-Weil group structure
\(\Z/{2}\Z \times \Z/{4}\Z\)
Torsion generators
\( \left(-6, 0\right) \), \( \left(-20, 126\right) \)
Integral points
\( \left(-38, 0\right) \), \((-20,\pm 126)\), \( \left(-6, 0\right) \), \( \left(43, 0\right) \), \((106,\pm 1008)\)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 336 \) | = | \(2^{4} \cdot 3 \cdot 7\) |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | \(258096513024 \) | = | \(2^{14} \cdot 3^{8} \cdot 7^{4} \) |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{124475734657}{63011844} \) | = | \(2^{-2} \cdot 3^{-8} \cdot 7^{-4} \cdot 4993^{3}\) |
Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | \(0.88224837397766581825870317593\dots\) | ||
Stable Faltings height: | \(0.18910119341772050884147105447\dots\) |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | \(0\) | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | \(1\) | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | \(0.78849338560790657106220152661\dots\) | ||
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: | \( 128 \) = \( 2^{2}\cdot2^{3}\cdot2^{2} \) | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | \(8\) | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | \(1\) (exact) |
Modular invariants

For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 384 | ||
\( \Gamma_0(N) \)-optimal: | no | ||
Manin constant: | 1 |
Special L-value
\( L(E,1) \) ≈ \( 1.5769867712158131421244030532285101726 \)
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(2\) | \(4\) | \(I_6^{*}\) | Additive | -1 | 4 | 14 | 2 |
\(3\) | \(8\) | \(I_{8}\) | Split multiplicative | -1 | 1 | 8 | 8 |
\(7\) | \(4\) | \(I_{4}\) | Split multiplicative | -1 | 1 | 4 | 4 |
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 X200a.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 2 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ 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 |
---|---|
\(2\) | Cs |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
$p$ | 2 | 3 | 7 |
---|---|---|---|
Reduction type | add | split | split |
$\lambda$-invariant(s) | - | 3 | 1 |
$\mu$-invariant(s) | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
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 336.d
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{4}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \times \Z/8\Z\) | 2.2.28.1-126.1-a4 |
$4$ | \(\Q(\zeta_{8})\) | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-7})\) | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
$8$ | 8.0.16777216.2 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
$8$ | 8.0.157351936.1 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
$8$ | 8.4.2498119335936.25 | \(\Z/2\Z \times \Z/16\Z\) | Not in database |
$8$ | 8.2.435537865728.11 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
$16$ | 16.0.1622647227216566419456.13 | \(\Z/8\Z \times \Z/8\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/2\Z \times \Z/16\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/2\Z \times \Z/24\Z\) | 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.