Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-9667701x-10337982173\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-9667701xz^2-10337982173z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-783083808x-7534039752720\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-\frac{384214311918474818}{285966599839209}, \frac{72962507621960701665047335}{4835851627011136237323}\right)\) |
$\hat{h}(P)$ | ≈ | $38.404197301345641505073620802$ |
Integral points
None
Invariants
Conductor: | \( 435344 \) | = | $2^{4} \cdot 7 \cdot 13^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $11688793608998391861248 $ | = | $2^{12} \cdot 7^{11} \cdot 13^{7} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{5054443262672896}{591220696157} \) | = | $2^{15} \cdot 7^{-11} \cdot 13^{-1} \cdot 23^{-1} \cdot 31^{3} \cdot 173^{3}$ | 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: | $2.9659892303409426883377437179\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.99036737105022901089376787566\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9436076427481541\dots$ | |||
Szpiro ratio: | $4.6108291203516965\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $38.404197301345641505073620802\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.086272306788922179736009546720\dots$ | 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\cdot1 $ | 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) $ ≈ $ 6.6264373831279768833627818851 $ | 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
$\displaystyle 6.626437383 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.086272 \cdot 38.404197 \cdot 2}{1^2} \approx 6.626437383$
Modular invariants
Modular form 435344.2.a.n
For more coefficients, see the Downloads section to the right.
Modular degree: | 50029056 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II^{*}$ | Additive | -1 | 4 | 12 | 0 |
$7$ | $1$ | $I_{11}$ | Non-split multiplicative | 1 | 1 | 11 | 11 |
$13$ | $2$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
$23$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 \( 4186 = 2 \cdot 7 \cdot 13 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1795 & 2 \\ 1795 & 3 \end{array}\right),\left(\begin{array}{rr} 3823 & 2 \\ 3823 & 3 \end{array}\right),\left(\begin{array}{rr} 1289 & 2 \\ 1289 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 4185 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 4185 & 2 \\ 4184 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[4186])$ is a degree-$42347726733312$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4186\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 435344.n consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 2093.i1, its twist by $-52$.
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.