Minimal Weierstrass equation
\(y^2+xy=x^3-x^2-231x-451\)
Mordell-Weil group structure
$\Z\times \Z/{2}\Z$
Infinite order Mordell-Weil generator and height
$P$ | = |
\(\left(25, 82\right)\)
|
$\hat{h}(P)$ | ≈ | $0.85208060646282640507989603419$ |
Torsion generators
\( \left(-2, 1\right) \)
Integral points
\( \left(-10, 33\right) \), \( \left(-10, -23\right) \), \( \left(-2, 1\right) \), \( \left(23, 66\right) \), \( \left(23, -89\right) \), \( \left(25, 82\right) \), \( \left(25, -107\right) \)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 1386 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 11$ |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | $678984768 $ | = | $2^{6} \cdot 3^{9} \cdot 7^{2} \cdot 11 $ |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{69426531}{34496} \) | = | $2^{-6} \cdot 3^{3} \cdot 7^{-2} \cdot 11^{-1} \cdot 137^{3}$ |
Endomorphism ring: | $\Z$ | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | $0.38764098207549041979939507112\dots$ | ||
Stable Faltings height: | $-0.43631823442559184874703885657\dots$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | $0.85208060646282640507989603419\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | $1.2887415247106829140646082139\dots$ | ||
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: | $ 8 $ = $ 2\cdot2\cdot2\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | $2$ | ||
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) $ ≈ $ 2.1962233198986125580980933303 $ |
Modular invariants
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 576 | ||
$ \Gamma_0(N) $-optimal: | yes | ||
Manin constant: | 1 |
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$3$ | $2$ | $III^{*}$ | Additive | 1 | 2 | 9 | 0 |
$7$ | $2$ | $I_{2}$ | Non-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 |
$p$-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 | nonsplit | add | ord | nonsplit | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | - | 1 | 1 | 3 | 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 |
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.
Its isogeny class 1386.d
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
$4$ | 4.0.19008.2 | \(\Z/4\Z\) | Not in database |
$8$ | 8.4.3175234143814656.26 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
$8$ | 8.0.43717791744.1 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
$8$ | 8.2.76879700667.1 | \(\Z/6\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | 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.