Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-30287023212x-2028728475306416\)
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(homogenize, simplify) |
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\(y^2z=x^3-30287023212xz^2-2028728475306416z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-30287023212x-2028728475306416\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Torsion generators
\( \left(-100852, 0\right) \), \( \left(200954, 0\right) \)
Integral points
\( \left(-100852, 0\right) \), \( \left(-100102, 0\right) \), \( \left(200954, 0\right) \)
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: | $74300685813099962302464000000 $ | = | $2^{30} \cdot 3^{20} \cdot 5^{6} \cdot 7^{4} \cdot 23^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{16077778198622525072705635801}{388799208512064000000} \) | = | $2^{-12} \cdot 3^{-14} \cdot 5^{-6} \cdot 7^{-4} \cdot 23^{-2} \cdot 631^{3} \cdot 3999871^{3}$ | 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: | $4.6501823374260412119522837127\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $3.0611554222520684021288129121\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0374083227231057\dots$ | |||
| Szpiro ratio: | $6.439478880038196\dots$ | |||
BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.011444215764683867770643606685\dots$ | 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: | $ 768 $ = $ 2^{2}\cdot2^{2}\cdot( 2 \cdot 3 )\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: | $4$ = $2^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L(E,1) $ ≈ $ 2.1972894268193026119635724835 $ | 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
$\displaystyle 2.197289427 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.011444 \cdot 1.000000 \cdot 768}{4^2} \approx 2.197289427$
Modular invariants
Modular form 463680.2.a.lc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 990904320 | 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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{20}^{*}$ | Additive | -1 | 6 | 30 | 12 |
| $3$ | $4$ | $I_{14}^{*}$ | Additive | -1 | 2 | 20 | 14 |
| $5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
| $23$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2757 & 4 \\ 2756 & 5 \end{array}\right),\left(\begin{array}{rr} 1383 & 2756 \\ 1384 & 2755 \end{array}\right),\left(\begin{array}{rr} 831 & 2 \\ 826 & 1379 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2067 & 2756 \\ 2758 & 2757 \end{array}\right),\left(\begin{array}{rr} 1379 & 0 \\ 0 & 2759 \end{array}\right),\left(\begin{array}{rr} 1201 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 461 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2760])$ is a degree-$196977623040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2760\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 463680.lc
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 4830.n2, its twist by $24$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.