Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-1549704042x+20115499352116\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-1549704042xz^2+20115499352116z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-24795264675x+1287367163270750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Torsion generators
\( \left(-44716, 22358\right) \), \( \left(15284, -7642\right) \)
Integral points
\( \left(-44716, 22358\right) \), \( \left(15284, -7642\right) \)
Invariants
Conductor: | \( 450450 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $63396648236188282640625000000 $ | = | $2^{6} \cdot 3^{10} \cdot 5^{12} \cdot 7^{6} \cdot 11^{2} \cdot 13^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{36134533748915083453404889}{5565686539253841000000} \) | = | $2^{-6} \cdot 3^{-4} \cdot 5^{-6} \cdot 7^{-6} \cdot 11^{-2} \cdot 13^{-6} \cdot 103^{3} \cdot 1753^{3} \cdot 1831^{3}$ | 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: | $4.2500798872155710117626230397\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.8960547866644659787646207546\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9941380459135555\dots$ | |||
Szpiro ratio: | $5.768749334294389\dots$ |
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: | $0.033471526799344540216505016194\dots$ | 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: | $ 256 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2\cdot2\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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 0.53554442878951264346408025911 $ | 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
$\displaystyle 0.535544429 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.033472 \cdot 1.000000 \cdot 256}{4^2} \approx 0.535544429$
Modular invariants
Modular form 450450.2.a.e
For more coefficients, see the Downloads section to the right.
Modular degree: | 573308928 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$5$ | $4$ | $I_{6}^{*}$ | Additive | 1 | 2 | 12 | 6 |
$7$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$11$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$13$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
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 \( 120120 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 120109 & 12 \\ 120108 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 48041 & 120114 \\ 96102 & 5 \end{array}\right),\left(\begin{array}{rr} 36961 & 12 \\ 101646 & 73 \end{array}\right),\left(\begin{array}{rr} 60067 & 6 \\ 120114 & 120115 \end{array}\right),\left(\begin{array}{rr} 51481 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 120104 & 120113 \end{array}\right),\left(\begin{array}{rr} 90097 & 6 \\ 120072 & 120079 \end{array}\right),\left(\begin{array}{rr} 76441 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 40039 & 120108 \\ 100100 & 120119 \end{array}\right)$.
The torsion field $K:=\Q(E[120120])$ is a degree-$64274810535936000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120120\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 450450e
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30030r6, its twist by $-15$.
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 $0$.