Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+x^2+39527x-30211604\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+x^2z+39527xz^2-30211604z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+51226560x-1410167305008\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(1052, 34303\right)\) | \(\left(326, 4174\right)\) |
$\hat{h}(P)$ | ≈ | $0.64539033141766840740143777137$ | $1.3615541602260914110596620578$ |
Integral points
\( \left(296, 2740\right) \), \( \left(296, -2741\right) \), \( \left(326, 4174\right) \), \( \left(326, -4175\right) \), \( \left(1052, 34303\right) \), \( \left(1052, -34304\right) \), \( \left(1196, 41584\right) \), \( \left(1196, -41585\right) \), \( \left(1430, 54337\right) \), \( \left(1430, -54338\right) \), \( \left(25442, 4058338\right) \), \( \left(25442, -4058339\right) \), \( \left(275480, 144589252\right) \), \( \left(275480, -144589253\right) \)
Invariants
Conductor: | \( 409101 \) | = | $3 \cdot 7^{2} \cdot 11^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-398600413559514243 $ | = | $-1 \cdot 3^{11} \cdot 7^{4} \cdot 11^{6} \cdot 23^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1605632000}{93710763} \) | = | $2^{18} \cdot 3^{-11} \cdot 5^{3} \cdot 7^{2} \cdot 23^{-2}$ | 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: | $2.0568757479697514610566878645\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.20929139521879508732393182771\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.121644944489585\dots$ | |||
Szpiro ratio: | $3.7124702198963258\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.87463643057451208052329709034\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.14323568270458494105339280301\dots$ | 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: | $ 132 $ = $ 11\cdot3\cdot2\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^{(2)}(E,1)/2! $ ≈ $ 16.536847305216684276664133075 $ | 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
$\displaystyle 16.536847305 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.143236 \cdot 0.874636 \cdot 132}{1^2} \approx 16.536847305$
Modular invariants
Modular form 409101.2.a.bf
For more coefficients, see the Downloads section to the right.
Modular degree: | 2956800 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $11$ | $I_{11}$ | Split multiplicative | -1 | 1 | 11 | 11 |
$7$ | $3$ | $IV$ | Additive | 1 | 2 | 4 | 0 |
$11$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$23$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
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 label 6.2.0.a.1, level \( 6 = 2 \cdot 3 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 5 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[6])$ is a degree-$144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 409101.bf consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 3381.h1, its twist by $-11$.
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.