Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+6615x+240786\) | (homogenize, simplify) |
\(y^2z=x^3+6615xz^2+240786z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+6615x+240786\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{1318}{9}, \frac{56314}{27}\right)\) |
$\hat{h}(P)$ | ≈ | $7.5260044602457970009598379777$ |
Integral points
None
Invariants
Conductor: | \( 508032 \) | = | $2^{7} \cdot 3^{4} \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-43571933983872 $ | = | $-1 \cdot 2^{7} \cdot 3^{10} \cdot 7^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{36000}{49} \) | = | $2^{5} \cdot 3^{2} \cdot 5^{3} \cdot 7^{-2}$ | 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: | $1.3028539379779725977608957519\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.98994723243307689478120372144\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.7183441483882939\dots$ | |||
Szpiro ratio: | $2.9156339911543276\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $7.5260044602457970009598379777\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.43266661775147656568407421366\dots$ | 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: | $ 2 $ = $ 1\cdot1\cdot2 $ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 6.5125017899941519239651500396 $ | 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
$\displaystyle 6.512501790 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.432667 \cdot 7.526004 \cdot 2}{1^2} \approx 6.512501790$
Modular invariants
Modular form 508032.2.a.bw
For more coefficients, see the Downloads section to the right.
Modular degree: | 940032 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II$ | Additive | 1 | 7 | 7 | 0 |
$3$ | $1$ | $IV^{*}$ | Additive | -1 | 4 | 10 | 0 |
$7$ | $2$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
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 |
---|---|---|
$2$ | 2G | 8.2.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 8.2.0.a.1, level \( 8 = 2^{3} \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 6 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 7 & 0 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 7 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[8])$ is a degree-$768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 508032.bw consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 72576.q1, its twist by $56$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.