Minimal Weierstrass equation
\( y^2 + x y + y = x^{3} - 86 x - 2456 \)
Mordell-Weil group structure
Integral points
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 38 \) | = | \(2 \cdot 19\) | ||
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-2550136832 \) | = | \(-1 \cdot 2^{27} \cdot 19 \) | ||
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{69173457625}{2550136832} \) | = | \(-1 \cdot 2^{-27} \cdot 5^{3} \cdot 19^{-1} \cdot 821^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no 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: | \(0\) | ||
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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: | \(0.630210743314\) | ||
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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: | \( 1 \) = \( 1\cdot1 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(1\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 38.2.a.a
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 18 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.630210743314 \)
Local data
This elliptic curve is semistable.
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \( I_{27} \) | Non-split multiplicative | 1 | 1 | 27 | 27 |
| \(19\) | \(1\) | \( I_{1} \) | 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.1.2 |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 | 19 |
|---|---|---|---|
| Reduction type | nonsplit | ordinary | split |
| $\lambda$-invariant(s) | 1 | 0 | 1 |
| $\mu$-invariant(s) | 0 | 2 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class 38a
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 2 | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-1444.2-b4 |
| 3 | 3.1.152.1 | \(\Z/2\Z\) | Not in database |
| 3.1.1083.1 | \(\Z/3\Z\) | Not in database | |
| 6 | 6.0.623808.1 | \(\Z/6\Z\) | Not in database |
| 6.0.3511808.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database | |
| 6.0.3518667.2 | \(\Z/3\Z \times \Z/3\Z\) | Not in database | |
| 6.0.7105563.1 | \(\Z/9\Z\) | Not in database | |
| 6.0.2565108243.1 | \(\Z/9\Z\) | 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.