Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-1191420x-500774400\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-1191420xz^2-500774400z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-96505047x-364775022486\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Torsion generators
\( \left(-640, 0\right) \), \( \left(-621, 0\right) \)
Integral points
\( \left(-640, 0\right) \), \( \left(-621, 0\right) \), \( \left(1260, 0\right) \)
Invariants
Conductor: | \( 476520 \) | = | $2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $73775468748960000 $ | = | $2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 11^{2} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{15529488955216}{6125625} \) | = | $2^{4} \cdot 3^{-4} \cdot 5^{-4} \cdot 11^{-2} \cdot 9901^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.2010340300793718022215814038\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.26671642012285469927224627355\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0278304524202422\dots$ | |||
Szpiro ratio: | $4.098553400286997\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.14450856346196563399702748584\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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Tamagawa product: | $ 256 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2\cdot2^{2} $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $4$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 2.3121370153914501439524397735 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 2.312137015 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.144509 \cdot 1.000000 \cdot 256}{4^2} \approx 2.312137015$
Modular invariants
Modular form 476520.2.a.cj
For more coefficients, see the Downloads section to the right.
Modular degree: | 7077888 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 4 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{1}^{*}$ | Additive | 1 | 3 | 8 | 0 |
$3$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$5$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$11$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$19$ | $4$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2Cs | 8.24.0.15 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8360 = 2^{3} \cdot 5 \cdot 11 \cdot 19 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 8353 & 8 \\ 8352 & 9 \end{array}\right),\left(\begin{array}{rr} 2205 & 3192 \\ 7942 & 6499 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 6385 & 1216 \\ 304 & 3269 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2639 & 0 \\ 0 & 8359 \end{array}\right),\left(\begin{array}{rr} 4447 & 5282 \\ 5662 & 875 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 8356 & 8357 \end{array}\right),\left(\begin{array}{rr} 5017 & 4408 \\ 5548 & 913 \end{array}\right)$.
The torsion field $K:=\Q(E[8360])$ is a degree-$6240706560000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8360\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 476520.cj
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1320.f4, its twist by $-19$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.