Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-33803211x-51308803910\) | (homogenize, simplify) |
\(y^2z=x^3-33803211xz^2-51308803910z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-33803211x-51308803910\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{41769}{4}, \frac{6855485}{8}\right)\) |
$\hat{h}(P)$ | ≈ | $8.3363849557151610945014230376$ |
Torsion generators
\( \left(-4810, 0\right) \), \( \left(6461, 0\right) \)
Integral points
\( \left(-4810, 0\right) \), \( \left(-1651, 0\right) \), \( \left(6461, 0\right) \)
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: | $1334750271935028818448384 $ | = | $2^{12} \cdot 3^{14} \cdot 13^{8} \cdot 17^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{296380748763217}{92608836489} \) | = | $3^{-8} \cdot 13^{-2} \cdot 17^{-4} \cdot 61^{3} \cdot 1093^{3}$ | 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: | $3.3335103586076737422948768154\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.80858235498290521915327835470\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9639015803583432\dots$ | |||
Szpiro ratio: | $4.919366681294589\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $8.3363849557151610945014230376\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.064146619841352158576793714694\dots$ | 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^{2}\cdot2^{2}\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: | $4$ | 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
$\displaystyle 4.278007333 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.064147 \cdot 8.336385 \cdot 128}{4^2} \approx 4.278007333$
Modular invariants
Modular form 413712.2.a.bb
For more coefficients, see the Downloads section to the right.
Modular degree: | 44040192 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 4 | 12 | 0 |
$3$ | $4$ | $I_{8}^{*}$ | Additive | -1 | 2 | 14 | 8 |
$13$ | $4$ | $I_{2}^{*}$ | Additive | 1 | 2 | 8 | 2 |
$17$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
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 |
---|---|---|
$2$ | 2Cs | 8.24.0.13 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5304 = 2^{3} \cdot 3 \cdot 13 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 3535 & 0 \\ 0 & 5303 \end{array}\right),\left(\begin{array}{rr} 1873 & 3540 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 889 & 444 \\ 2232 & 4429 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5297 & 1320 \\ 6 & 3983 \end{array}\right),\left(\begin{array}{rr} 5297 & 8 \\ 5296 & 9 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 5300 & 5301 \end{array}\right),\left(\begin{array}{rr} 3665 & 3534 \\ 426 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[5304])$ is a degree-$788363476992$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5304\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 413712.bb
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 663.a2, 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.