Minimal Weierstrass equation
\(y^2=x^3+x^2-13x-4\)
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(-1, 3\right) \)
Integral points
\( \left(-4, 0\right) \), \((-1,\pm 3)\), \((5,\pm 9)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 156 \) | = | \(2^{2} \cdot 3 \cdot 13\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(151632 \) | = | \(2^{4} \cdot 3^{6} \cdot 13 \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{16384000}{9477} \) | = | \(2^{17} \cdot 3^{-6} \cdot 5^{3} \cdot 13^{-1}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(0\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(2.7522991872794050113370038521\) | ||
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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: | \( 18 \) = \( 3\cdot( 2 \cdot 3 )\cdot1 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(6\) | ||
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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.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 12 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 1.3761495936397025056685019260682219193 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(3\) | \(IV\) | Additive | -1 | 2 | 4 | 0 |
| \(3\) | \(6\) | \(I_{6}\) | Split multiplicative | -1 | 1 | 6 | 6 |
| \(13\) | \(1\) | \(I_{1}\) | 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 X6.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.
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 |
| \(3\) | B.1.1 |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 | 13 |
|---|---|---|---|
| Reduction type | add | split | split |
| $\lambda$-invariant(s) | - | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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, 3 and 6.
Its isogeny class 156.b
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \times \Z/6\Z\) | 2.2.13.1-1872.1-h3 |
| $4$ | 4.0.7488.1 | \(\Z/12\Z\) | Not in database |
| $6$ | 6.0.12338352.2 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $8$ | 8.4.177935486976.6 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $8$ | 8.0.9475854336.3 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $9$ | 9.3.26228424500928.2 | \(\Z/18\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/6\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/24\Z\) | Not in database |
| $18$ | 18.6.255423686981904214279152247590912.1 | \(\Z/2\Z \times \Z/18\Z\) | 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.