Minimal Weierstrass equation
Minimal equation
Minimal equation
Simplified equation
\(y^2=x^3+x^2-526485408x-4649899660812\)
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(homogenize, simplify) |
\(y^2z=x^3+x^2z-526485408xz^2-4649899660812z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-42645318075x-3389648916777750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = |
\(\left(\frac{8454888491563501299708}{152073658200615529}, \frac{696499942684697738692175964198726}{59303590714581646921739083}\right)\)
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$\hat{h}(P)$ | ≈ | $49.460302584238168610812714310$ |
Torsion generators
\( \left(-13237, 0\right) \)
Integral points
\( \left(-13237, 0\right) \)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 433200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 19^{2}$ |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | $18586811392070400000000 $ | = | $2^{14} \cdot 3^{2} \cdot 5^{8} \cdot 19^{9} $ |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{781484460931}{900} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-2} \cdot 61^{3} \cdot 151^{3}$ |
Endomorphism ring: | $\Z$ | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | $3.5570851496119131007982368808\dots$ | ||
Stable Faltings height: | $-0.14911022153991274092614548119\dots$ |
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: | $49.460302584238168610812714310\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | $0.031517589873179802110961370599\dots$ | ||
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: | $ 32 $ = $ 2\cdot2\cdot2^{2}\cdot2 $ | ||
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) $ ≈ $ 12.470956254827149593933211512 $ |
Modular invariants
Modular form 433200.2.a.jp
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 84049920 | ||
$ \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)_{-}$ |
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$2$ | $2$ | $I_{6}^{*}$ | Additive | -1 | 4 | 14 | 2 |
$3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$5$ | $4$ | $I_{2}^{*}$ | Additive | 1 | 2 | 8 | 2 |
$19$ | $2$ | $III^{*}$ | Additive | 1 | 2 | 9 | 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 |
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$2$ | 2B | 2.3.0.1 |
$p$-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 433200.jp
consists of 2 curves linked by isogenies of
degree 2.