Minimal Weierstrass equation
\( y^2 + y = x^{3} + x^{2} - x - 35 \)
Mordell-Weil group structure
\(\Z\times \Z/{3}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(11, 37\right) \) |
| \(\hat{h}(P)\) | ≈ | $1.8702142681361649$ |
Torsion generators
\( \left(5, 10\right) \)
Integral points
\( \left(5, 10\right) \), \( \left(5, -11\right) \), \( \left(11, 37\right) \), \( \left(11, -38\right) \), \( \left(47, 325\right) \), \( \left(47, -326\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 1155 \) | = | \(3 \cdot 5 \cdot 7 \cdot 11\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-509355 \) | = | \(-1 \cdot 3^{3} \cdot 5 \cdot 7^{3} \cdot 11 \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{262144}{509355} \) | = | \(-1 \cdot 2^{18} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-3} \cdot 11^{-1}\) |
| 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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| Rank: | \(1\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1.87021426814\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(1.32968082968\) | ||
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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: | \( 9 \) = \( 3\cdot1\cdot3\cdot1 \) | ||
|
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(3\) | ||
|
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: | 144 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 2.48678805974 \)
Local data
This elliptic curve is semistable.
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(3\) | \( I_{3} \) | Split multiplicative | -1 | 1 | 3 | 3 |
| \(5\) | \(1\) | \( I_{1} \) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(7\) | \(3\) | \( I_{3} \) | Split multiplicative | -1 | 1 | 3 | 3 |
| \(11\) | \(1\) | \( I_{1} \) | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
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.1.1 |
$p$-adic data
$p$-adic regulators
Note: \(p\)-adic regulator data only exists for primes \(p\ge 5\) of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | split | nonsplit | split | nonsplit | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | 8,1 | 2 | 1 | 2 | 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 |
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class 1155.h
consists of 2 curves linked by isogenies of
degree 3.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.4620.1 | \(\Z/6\Z\) | Not in database |
| $6$ | 6.0.247066875.1 | \(\Z/3\Z \times \Z/3\Z\) | Not in database |
| $6$ | 6.0.24652782000.1 | \(\Z/2\Z \times \Z/6\Z\) | 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.