Minimal Weierstrass equation
\(y^2+xy=x^3-363x+1305\)
Mordell-Weil group structure
$\Z/{6}\Z$
Torsion generators
\( \left(-12, 69\right) \)
Integral points
\( \left(-12, 69\right) \), \( \left(-12, -57\right) \), \( \left(24, 69\right) \), \( \left(24, -93\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 462 \) | = | $2 \cdot 3 \cdot 7 \cdot 11$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $2291573592 $ | = | $2^{3} \cdot 3^{12} \cdot 7^{2} \cdot 11 $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{5290763640625}{2291573592} \) | = | $2^{-3} \cdot 3^{-12} \cdot 5^{6} \cdot 7^{-2} \cdot 11^{-1} \cdot 17^{3} \cdot 41^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.49225057390419977497426756934\dots$ | ||
| Stable Faltings height: | $0.49225057390419977497426756934\dots$ | ||
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: | $1.3136031374174197867722777393\dots$ | ||
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: | $ 72 $ = $ 3\cdot( 2^{2} \cdot 3 )\cdot2\cdot1 $ | ||
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) $ ≈ $ 2.6272062748348395735445554786 $ | ||
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: | 192 | ||
| $ \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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
| $7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
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 | 2.3.0.1 |
| $3$ | 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 | 7 | 11 |
|---|---|---|---|---|
| Reduction type | split | split | split | nonsplit |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 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 $d=$
2, 3 and 6.
Its isogeny class 462.f
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{22}) \) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $4$ | 4.0.17248.1 | \(\Z/12\Z\) | Not in database |
| $6$ | 6.0.949132107.1 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $8$ | 8.4.364092804038656.4 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $8$ | 8.0.2303789694976.17 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $9$ | 9.3.32282093828875968.13 | \(\Z/18\Z\) | 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 |
| $18$ | 18.6.351978977777939262204503421894180265732866048.2 | \(\Z/2\Z \times \Z/18\Z\) | 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.