Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+94325829x-347466266966\)
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(homogenize, simplify) |
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\(y^2z=x^3+94325829xz^2-347466266966z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+94325829x-347466266966\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{123882993}{14884}, \frac{1828723437905}{1815848}\right)\)
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| $\hat{h}(P)$ | ≈ | $16.672769911430322189002846075$ |
Torsion generators
\( \left(3302, 0\right) \)
Integral points
\( \left(3302, 0\right) \)
Invariants
| Conductor: | \( 413712 \) | = | $2^{4} \cdot 3^{2} \cdot 13^{2} \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-105868639565323401657765888 $ | = | $-1 \cdot 2^{12} \cdot 3^{10} \cdot 13^{7} \cdot 17^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{6439735268725823}{7345472585373} \) | = | $3^{-4} \cdot 13^{-1} \cdot 17^{-8} \cdot 23^{3} \cdot 8089^{3}$ | 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: | $3.6800839488876463970034928761\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.1551559452628778738618944154\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.9885418782056836\dots$ | |||
| Szpiro ratio: | $5.1574101002767625\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $16.672769911430322189002846075\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.032073309920676079288396857347\dots$ | 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\cdot2\cdot2^{2}\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: | $2$ | 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) $ ≈ $ 4.2780073328434223077653235556 $ | 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
$\displaystyle 4.278007333 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.032073 \cdot 16.672770 \cdot 32}{2^2} \approx 4.278007333$
Modular invariants
Modular form 413712.2.a.bb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 88080384 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}^{*}$ | Additive | -1 | 4 | 12 | 0 |
| $3$ | $2$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
| $13$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
| $17$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
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 |
|---|---|---|
| $2$ | 2B | 8.24.0.91 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10608 = 2^{4} \cdot 3 \cdot 13 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 10593 & 16 \\ 10592 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 3552 \\ 4164 & 4105 \end{array}\right),\left(\begin{array}{rr} 3535 & 0 \\ 0 & 10607 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 10604 & 10605 \end{array}\right),\left(\begin{array}{rr} 2168 & 7071 \\ 3729 & 10598 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 10510 & 10595 \end{array}\right),\left(\begin{array}{rr} 434 & 9723 \\ 3015 & 5294 \end{array}\right),\left(\begin{array}{rr} 1873 & 3552 \\ 840 & 7201 \end{array}\right)$.
The torsion field $K:=\Q(E[10608])$ is a degree-$12613815631872$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10608\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 413712bb
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 663b6, its twist by $156$.
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.