Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2+7414840x+75441366900\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z+7414840xz^2+75441366900z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+600602013x+54994954664034\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(46980, 10203750\right)\) |
$\hat{h}(P)$ | ≈ | $5.2940624949896469842795416179$ |
Torsion generators
\( \left(-3645, 0\right) \)
Integral points
\( \left(-3645, 0\right) \), \((46980,\pm 10203750)\)
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: | $-2484614302500000000000000 $ | = | $-1 \cdot 2^{14} \cdot 3 \cdot 5^{16} \cdot 11^{7} \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{6213165856218719}{342407226562500} \) | = | $2^{-2} \cdot 3^{-1} \cdot 5^{-16} \cdot 11^{-1} \cdot 17^{-1} \cdot 23^{3} \cdot 7993^{3}$ | 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: | $3.3606889543494031858560709475\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.4685941373902726044078670371\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9930120230079293\dots$ | |||
Szpiro ratio: | $4.852635888574718\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $5.2940624949896469842795416179\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.061916876267413208331498662184\dots$ | 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: | $ 128 $ = $ 2\cdot1\cdot2^{4}\cdot2^{2}\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: | $2$ | 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
$\displaystyle 10.489337999 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.061917 \cdot 5.294062 \cdot 128}{2^2} \approx 10.489337999$
Modular invariants
Modular form 493680.2.a.go
For more coefficients, see the Downloads section to the right.
Modular degree: | 70778880 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}^{*}$ | Additive | -1 | 4 | 14 | 2 |
$3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$5$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
$11$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$17$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
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 | 16.24.0.5 |
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
$\left(\begin{array}{rr} 8961 & 16 \\ 8960 & 17 \end{array}\right),\left(\begin{array}{rr} 8456 & 1 \\ 2719 & 10 \end{array}\right),\left(\begin{array}{rr} 3008 & 5 \\ 8931 & 8962 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 2508 & 2569 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 7846 & 2233 \\ 1205 & 4602 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 8972 & 8973 \end{array}\right),\left(\begin{array}{rr} 2432 & 8971 \\ 4941 & 14 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 8878 & 8963 \end{array}\right)$.
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 5610bb4, 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.