Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
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\(y^2+xy=x^3-41608x-90515392\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-41608xz^2-90515392z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-53923995x-4222924357194\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(2864, 151160\right) \)
Integral points
\( \left(752, 17048\right) \), \( \left(752, -17800\right) \), \( \left(2864, 151160\right) \), \( \left(2864, -154024\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 1122 \) | = | $2 \cdot 3 \cdot 11 \cdot 17$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-3534510366354604032 $ | = | $-1 \cdot 2^{15} \cdot 3^{6} \cdot 11^{6} \cdot 17^{4} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{7966267523043306625}{3534510366354604032} \) | = | $-1 \cdot 2^{-15} \cdot 3^{-6} \cdot 5^{3} \cdot 11^{-6} \cdot 17^{-4} \cdot 19^{3} \cdot 21023^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $2.2382123306049921058208225523\dots$ | ||
| Stable Faltings height: | $2.2382123306049921058208225523\dots$ | ||
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: | $0.11220167125559804079938946905\dots$ | ||
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: | $ 1080 $ = $ ( 3 \cdot 5 )\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $6$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L(E,1) $ ≈ $ 3.3660501376679412239816840715 $ | ||
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: | 23040 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $15$ | $I_{15}$ | Split multiplicative | -1 | 1 | 15 | 15 |
| $3$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $11$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $17$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 8.6.0.5 |
| $3$ | 3B.1.1 | 3.8.0.1 |
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 11 | 17 |
|---|---|---|---|---|---|
| Reduction type | split | split | ss | split | nonsplit |
| $\lambda$-invariant(s) | 6 | 3 | 2,2 | 1 | 0 |
| $\mu$-invariant(s) | 1 | 0 | 0,0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 1122.j
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $4$ | 4.2.34848.2 | \(\Z/12\Z\) | Not in database |
| $6$ | 6.0.2255067.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
| $8$ | 8.0.77720518656.13 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $8$ | 8.0.4567597056.5 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $9$ | 9.3.135878452228432951488.2 | \(\Z/18\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/24\Z\) | Not in database |
| $18$ | 18.0.38719620445623100574568675231701844967594917888.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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.