Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
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\(y^2=x^3+x^2-29080x+1347828\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-29080xz^2+1347828z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2355507x+989633106\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
| $P$ | = |
\(\left(-4, 1210\right)\)
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\(\left(-169, 1210\right)\)
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| $\hat{h}(P)$ | ≈ | $0.90071563153644120578487332205$ | $1.9979690663069686903267678746$ |
Torsion generators
\( \left(51, 0\right) \)
Integral points
\((-169,\pm 1210)\), \((-124,\pm 1750)\), \((-12,\pm 1302)\), \((-4,\pm 1210)\), \( \left(51, 0\right) \), \((151,\pm 650)\), \((172,\pm 1210)\), \((436,\pm 8470)\), \((98611,\pm 30966320)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 474320 \) | = | $2^{4} \cdot 5 \cdot 7^{2} \cdot 11^{2}$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $777786141440000 $ | = | $2^{11} \cdot 5^{4} \cdot 7^{3} \cdot 11^{6} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{2185454}{625} \) | = | $2 \cdot 5^{-4} \cdot 103^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $1.5638428833105486512076975289\dots$ | ||
| Stable Faltings height: | $-0.75696720586574814739874189061\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $2$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $1.6905885918118215260560467230\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.46921138317369852570968280429\dots$ | ||
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: | $ 64 $ = $ 2\cdot2^{2}\cdot2\cdot2^{2} $ | ||
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: | $1$ (rounded) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 12.691894584667199999508005565 $ | ||
Modular invariants
Modular form 474320.2.a.br
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: | 1966080 | ||
| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | ||
| Manin constant: | 1 (conditional*) | ||
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | Additive | 1 | 4 | 11 | 0 |
| $5$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
| $11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
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 | 8.6.0.3 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 474320.br
consists of 2 curves linked by isogenies of
degree 2.