Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-120914819x-502311261058\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-120914819xz^2-502311261058z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-156705604803x-23435364079096002\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-5844, 71749\right)\) |
$\hat{h}(P)$ | ≈ | $3.8243512577781672939066873782$ |
Torsion generators
\( \left(-7041, 3520\right) \), \( \left(12671, -6336\right) \)
Integral points
\( \left(-7041, 3520\right) \), \( \left(-5844, 71749\right) \), \( \left(-5844, -65906\right) \), \( \left(12671, -6336\right) \), \( \left(16191, 1327744\right) \), \( \left(16191, -1343936\right) \), \( \left(115158, 38836237\right) \), \( \left(115158, -38951396\right) \)
Invariants
Conductor: | \( 482790 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $4144221779577974128742400 $ | = | $2^{12} \cdot 3^{6} \cdot 5^{2} \cdot 7^{2} \cdot 11^{12} \cdot 19^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{110358600993178429667329}{2339305154932838400} \) | = | $2^{-12} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{-6} \cdot 19^{-2} \cdot 97^{3} \cdot 494497^{3}$ | 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: | $3.5129712467960394864977541465\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.3140236103968542144667823575\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9666984951419172\dots$ | |||
Szpiro ratio: | $5.153484882085558\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.8243512577781672939066873782\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.045586850585840054394169903504\dots$ | 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: | $ 384 $ = $ 2\cdot( 2 \cdot 3 )\cdot2\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)
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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) $ ≈ $ 4.1841631050264670699536884456 $ | 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
$\displaystyle 4.184163105 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.045587 \cdot 3.824351 \cdot 384}{4^2} \approx 4.184163105$
Modular invariants
Modular form 482790.2.a.cc
For more coefficients, see the Downloads section to the right.
Modular degree: | 119439360 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 6 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$3$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$11$ | $4$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
$19$ | $2$ | $I_{2}$ | 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 |
---|---|---|
$2$ | 2Cs | 2.6.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 87780 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 17557 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 87764 & 87773 \end{array}\right),\left(\begin{array}{rr} 43897 & 12 \\ 87714 & 87667 \end{array}\right),\left(\begin{array}{rr} 87769 & 12 \\ 87768 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 58527 & 10 \\ 58522 & 3 \end{array}\right),\left(\begin{array}{rr} 15959 & 87774 \\ 0 & 87779 \end{array}\right),\left(\begin{array}{rr} 37621 & 12 \\ 50166 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 69307 & 6 \\ 78534 & 87775 \end{array}\right)$.
The torsion field $K:=\Q(E[87780])$ is a degree-$18871896637440000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/87780\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 482790.cc
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 43890.ct6, its twist by $-11$.
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.