Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-3092558475x-65742737143750\) | (homogenize, simplify) |
\(y^2z=x^3-3092558475xz^2-65742737143750z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-3092558475x-65742737143750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-34150, 204050\right)\) |
$\hat{h}(P)$ | ≈ | $6.8663310670634071973872149165$ |
Torsion generators
\( \left(-34250, 0\right) \), \( \left(64165, 0\right) \)
Integral points
\( \left(-34250, 0\right) \), \((-34150,\pm 204050)\), \( \left(-29915, 0\right) \), \( \left(64165, 0\right) \)
Invariants
Conductor: | \( 428400 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $25775992539248627257344000000 $ | = | $2^{18} \cdot 3^{22} \cdot 5^{6} \cdot 7^{4} \cdot 17^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{70108386184777836280897}{552468975892674624} \) | = | $2^{-6} \cdot 3^{-16} \cdot 7^{-4} \cdot 17^{-4} \cdot 41234113^{3}$ | 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: | $4.2797276601540954502414254669\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.2325553790430451078261910604\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.07814467781823\dots$ | |||
Szpiro ratio: | $5.950918556639019\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $6.8663310670634071973872149165\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.020254729454871682313180763625\dots$ | 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: | $ 1024 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2}\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: | $4$ | 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) $ ≈ $ 8.9008433991007809532234998791 $ | 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
$\displaystyle 8.900843399 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.020255 \cdot 6.866331 \cdot 1024}{4^2} \approx 8.900843399$
Modular invariants
Modular form 428400.2.a.nu
For more coefficients, see the Downloads section to the right.
Modular degree: | 377487360 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{10}^{*}$ | Additive | -1 | 4 | 18 | 6 |
$3$ | $4$ | $I_{16}^{*}$ | Additive | -1 | 2 | 22 | 16 |
$5$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$7$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$17$ | $4$ | $I_{4}$ | 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 |
---|---|---|
$2$ | 2Cs | 8.48.0.96 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 2036 & 2037 \end{array}\right),\left(\begin{array}{rr} 1223 & 0 \\ 0 & 2039 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 241 & 960 \\ 420 & 1801 \end{array}\right),\left(\begin{array}{rr} 2033 & 8 \\ 2032 & 9 \end{array}\right),\left(\begin{array}{rr} 679 & 0 \\ 0 & 2039 \end{array}\right),\left(\begin{array}{rr} 809 & 1320 \\ 930 & 1919 \end{array}\right),\left(\begin{array}{rr} 1771 & 2010 \\ 1050 & 811 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$14438891520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 428400nu
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 714g3, its twist by $60$.
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.