Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-46668x-3665072\) | (homogenize, simplify) |
\(y^2z=x^3-46668xz^2-3665072z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-46668x-3665072\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(266, 1656\right)\) | \(\left(381, 5819\right)\) |
$\hat{h}(P)$ | ≈ | $0.95424762294689044314024912858$ | $3.0876222597858716561501602329$ |
Torsion generators
\( \left(-148, 0\right) \)
Integral points
\( \left(-148, 0\right) \), \((-139,\pm 369)\), \((-112,\pm 396)\), \((-102,\pm 184)\), \((252,\pm 760)\), \((266,\pm 1656)\), \((381,\pm 5819)\), \((1968,\pm 86756)\), \((3578,\pm 213624)\)
Invariants
Conductor: | \( 463680 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $701904628776960 $ | = | $2^{15} \cdot 3^{7} \cdot 5 \cdot 7 \cdot 23^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{470547874952}{29383305} \) | = | $2^{3} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-1} \cdot 23^{-4} \cdot 3889^{3}$ | 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: | $1.6000738787629119655881948178\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.18433375872892548311903204752\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8524401601750782\dots$ | |||
Szpiro ratio: | $3.3621685675674815\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.7719726157958751577468738934\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.32609215977718624169963043670\dots$ | 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: | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot1\cdot2^{2} $ | 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^{(2)}(E,1)/2! $ ≈ $ 14.462696594049494605581336516 $ | 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
$\displaystyle 14.462696594 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.326092 \cdot 2.771973 \cdot 64}{2^2} \approx 14.462696594$
Modular invariants
Modular form 463680.2.a.s
For more coefficients, see the Downloads section to the right.
Modular degree: | 2097152 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{5}^{*}$ | Additive | -1 | 6 | 15 | 0 |
$3$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$5$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$23$ | $4$ | $I_{4}$ | 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 |
---|---|---|
$2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19320 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 19313 & 8 \\ 19312 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2768 & 3 \\ 13805 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 6436 & 19319 \\ 12857 & 19314 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 19314 & 19315 \end{array}\right),\left(\begin{array}{rr} 3868 & 1 \\ 7751 & 6 \end{array}\right),\left(\begin{array}{rr} 12071 & 12072 \\ 2402 & 12065 \end{array}\right),\left(\begin{array}{rr} 6721 & 8 \\ 7564 & 33 \end{array}\right),\left(\begin{array}{rr} 12072 & 2407 \\ 7253 & 7266 \end{array}\right)$.
The torsion field $K:=\Q(E[19320])$ is a degree-$397106888048640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 463680s
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 77280ba3, its twist by $-24$.
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.