Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-98934008x-397434693488\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-98934008xz^2-397434693488z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-8013654675x-289753932516750\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 473200 \) | = | $2^{4} \cdot 5^{2} \cdot 7 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-6272561379129500000000000 $ | = | $-1 \cdot 2^{11} \cdot 5^{12} \cdot 7 \cdot 13^{11} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{693346671296498}{40610171875} \) | = | $-1 \cdot 2 \cdot 5^{-6} \cdot 7^{-1} \cdot 13^{-5} \cdot 70249^{3}$ | 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: | $3.5142003924423564682131813477\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.79162184198125471258692851563\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.936308815989563\dots$ | |||
Szpiro ratio: | $5.1227139976359926\dots$ |
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: | $0.023853963727198812361729339660\dots$ | 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: | $ 32 $ = $ 2^{2}\cdot2\cdot1\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: | $1$ | 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: | $4$ = $2^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 3.0533073570814479823013554764 $ | 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
$\displaystyle 3.053307357 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.023854 \cdot 1.000000 \cdot 32}{1^2} \approx 3.053307357$
Modular invariants
Modular form 473200.2.a.dd
For more coefficients, see the Downloads section to the right.
Modular degree: | 100638720 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{3}^{*}$ | Additive | 1 | 4 | 11 | 0 |
$5$ | $2$ | $I_{6}^{*}$ | Additive | 1 | 2 | 12 | 6 |
$7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$13$ | $4$ | $I_{5}^{*}$ | Additive | 1 | 2 | 11 | 5 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 728 = 2^{3} \cdot 7 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 183 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 561 & 2 \\ 561 & 3 \end{array}\right),\left(\begin{array}{rr} 521 & 2 \\ 521 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 727 & 2 \\ 726 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 727 & 0 \end{array}\right),\left(\begin{array}{rr} 365 & 2 \\ 365 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[728])$ is a degree-$40577531904$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/728\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 473200dd consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 3640a1, its twist by $-260$.
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 $0$.