Minimal Weierstrass equation
\(y^2=x^3-155115x+23514138\)
Mordell-Weil group structure
trivial
Integral points
None
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 1296 \) | = | \(2^{4} \cdot 3^{4}\) |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | \(-278628139008 \) | = | \(-1 \cdot 2^{19} \cdot 3^{12} \) |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( -\frac{189613868625}{128} \) | = | \(-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 383^{3}\) |
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: | \(0\) | ||
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: | \(0.80848602428825605028754866728\) | ||
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: | \( 2 \) = \( 2\cdot1 \) | ||
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
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 3024 | ||
\( \Gamma_0(N) \)-optimal: | no | ||
Manin constant: | 1 |
Special L-value
\( L(E,1) \) ≈ \( 1.6169720485765121005750973345518439593 \)
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(2\) | \(2\) | \(I_{11}^{*}\) | Additive | -1 | 4 | 19 | 7 |
\(3\) | \(1\) | \(II^{*}\) | Additive | 1 | 4 | 12 | 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 X4.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.
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 |
\(7\) | B |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | add | ss | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
$\lambda$-invariant(s) | - | - | 0,0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
$\mu$-invariant(s) | - | - | 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 \(d=\)
3, 7 and 21.
Its isogeny class 1296.g
consists of 4 curves linked by isogenies of
degrees dividing 21.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{3}) \) | \(\Z/3\Z\) | 2.2.12.1-1458.1-i1 |
$3$ | 3.1.648.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.3359232.4 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
$6$ | 6.0.186624.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.7057326528.2 | \(\Z/7\Z\) | Not in database |
$6$ | 6.2.20155392.5 | \(\Z/6\Z\) | Not in database |
$12$ | 12.2.5777633090469888.3 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.313456656384.1 | \(\Z/3\Z \times \Z/3\Z\) | Not in database |
$12$ | 12.0.6499837226778624.48 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
$12$ | 12.12.448252719505312813056.1 | \(\Z/21\Z\) | Not in database |
$18$ | 18.6.1376809511370776442839236608.2 | \(\Z/9\Z\) | Not in database |
$18$ | 18.0.3537182715531733726396416.1 | \(\Z/6\Z\) | Not in database |
$18$ | 18.6.1439728439119866152968380211003392.1 | \(\Z/14\Z\) | Not in database |
We only show fields where the torsion growth is primitive.