Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2+101368920x-432451296972\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z+101368920xz^2-432451296972z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+8210882493x-315281628140094\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(8598, 1036728\right)\) |
$\hat{h}(P)$ | ≈ | $2.6470312474948234921397708089$ |
Torsion generators
\( \left(3747, 0\right) \)
Integral points
\( \left(3747, 0\right) \), \((8598,\pm 1036728)\)
Invariants
Conductor: | \( 493680 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-147467196867042749253427200 $ | = | $-1 \cdot 2^{13} \cdot 3^{8} \cdot 5^{2} \cdot 11^{14} \cdot 17^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{15875306080318016639}{20322604533582450} \) | = | $2^{-1} \cdot 3^{-8} \cdot 5^{-2} \cdot 11^{-8} \cdot 17^{-2} \cdot 23^{6} \cdot 4751^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $3.7072625446293758405646870083\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.8151677276702452591164830979\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.6470312474948234921397708089\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.030958438133706604165749331092\dots$ | 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);
|
Tamagawa product: | $ 512 $ = $ 2^{2}\cdot2^{3}\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)
|
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) $ ≈ $ 10.489337998535258576065612453 $ | 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
$\displaystyle 10.489337999 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.030958 \cdot 2.647031 \cdot 512}{2^2} \approx 10.489337999$
Modular invariants
Modular form 493680.2.a.go
For more coefficients, see the Downloads section to the right.
Modular degree: | 141557760 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{5}^{*}$ | Additive | -1 | 4 | 13 | 1 |
$3$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$11$ | $4$ | $I_{8}^{*}$ | Additive | -1 | 2 | 14 | 8 |
$17$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 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 |
---|---|---|
$2$ | 2B | 8.24.0.99 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8976 = 2^{4} \cdot 3 \cdot 11 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 4079 & 8960 \\ 5704 & 8847 \end{array}\right),\left(\begin{array}{rr} 2993 & 16 \\ 5992 & 129 \end{array}\right),\left(\begin{array}{rr} 8961 & 16 \\ 8960 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4765 & 16 \\ 7664 & 8661 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 8972 & 8973 \end{array}\right),\left(\begin{array}{rr} 2260 & 2245 \\ 4647 & 4498 \end{array}\right),\left(\begin{array}{rr} 5602 & 6731 \\ 3287 & 4478 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 8878 & 8963 \end{array}\right)$.
The torsion field $K:=\Q(E[8976])$ is a degree-$6353112268800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8976\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 493680go
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 5610bb6, its twist by $44$.
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.