Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
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\(y^2=x^3+x^2-337x+2015\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-337xz^2+2015z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-27324x+1550880\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-5, 60\right)\)
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| $\hat{h}(P)$ | ≈ | $1.8635922269785663635463051998$ |
Torsion generators
\( \left(13, 0\right) \)
Integral points
\((-5,\pm 60)\), \( \left(13, 0\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 11424 \) | = | $2^{5} \cdot 3 \cdot 7 \cdot 17$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $501559296 $ | = | $2^{12} \cdot 3 \cdot 7^{4} \cdot 17 $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{1036433728}{122451} \) | = | $2^{6} \cdot 3^{-1} \cdot 7^{-4} \cdot 11^{3} \cdot 17^{-1} \cdot 23^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.40071492901741566084245512646\dots$ | ||
| Stable Faltings height: | $-0.29243225154252964857477699500\dots$ | ||
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: | $1.8635922269785663635463051998\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $1.5987475376714832195669012260\dots$ | ||
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^{2}\cdot1\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) $ ≈ $ 5.9588269682113976680842422554 $ | ||
Modular invariants
Modular form 11424.2.a.s
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: | 3584 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | Additive | -1 | 5 | 12 | 0 |
| $3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
| $17$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | ord | nonsplit | ss | ord | nonsplit | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 4 | 1 | 3 | 1,1 | 3 | 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 | 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 and 4.
Its isogeny class 11424r
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{51}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.1153190317326336.8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | 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.