Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-4704300480x+47567535954228\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-4704300480xz^2+47567535954228z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-381048338907x+34677876855648906\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-28209, 12562830\right)\) |
$\hat{h}(P)$ | ≈ | $4.4676206917888195831191087233$ |
Torsion generators
\( \left(10347, 0\right) \), \( \left(62826, 0\right) \)
Integral points
\( \left(-73174, 0\right) \), \((-28209,\pm 12562830)\), \((-15574,\pm 10819200)\), \( \left(10347, 0\right) \), \( \left(62826, 0\right) \)
Invariants
Conductor: | \( 485520 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $5685390142070145334281216000000 $ | = | $2^{16} \cdot 3^{4} \cdot 5^{6} \cdot 7^{6} \cdot 17^{12} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{116454264690812369959009}{57505157319440250000} \) | = | $2^{-4} \cdot 3^{-4} \cdot 5^{-6} \cdot 7^{-6} \cdot 17^{-6} \cdot 73^{3} \cdot 307^{3} \cdot 2179^{3}$ | 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: | $4.5943974366856115607669950359\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.4846435840975582112249956055\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0316873731202008\dots$ | |||
Szpiro ratio: | $5.990145650360836\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $4.4676206917888195831191087233\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.021314809243128908133502426472\dots$ | 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: | $ 2304 $ = $ 2^{2}\cdot2^{2}\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\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) $ ≈ $ 13.712613525523338983841873809 $ | 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
$\displaystyle 13.712613526 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.021315 \cdot 4.467621 \cdot 2304}{4^2} \approx 13.712613526$
Modular invariants
Modular form 485520.2.a.ig
For more coefficients, see the Downloads section to the right.
Modular degree: | 764411904 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 6 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{8}^{*}$ | Additive | -1 | 4 | 16 | 4 |
$3$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$7$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$17$ | $4$ | $I_{6}^{*}$ | Additive | 1 | 2 | 12 | 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 | 4.12.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7140 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 9 & 4 \\ 7124 & 7133 \end{array}\right),\left(\begin{array}{rr} 2381 & 12 \\ 2380 & 1 \end{array}\right),\left(\begin{array}{rr} 7129 & 12 \\ 7128 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 5879 & 7134 \\ 0 & 7139 \end{array}\right),\left(\begin{array}{rr} 5719 & 6 \\ 4278 & 7135 \end{array}\right),\left(\begin{array}{rr} 3569 & 7128 \\ 7134 & 7067 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6121 & 12 \\ 1026 & 73 \end{array}\right)$.
The torsion field $K:=\Q(E[7140])$ is a degree-$909650165760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7140\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 485520ig
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 3570v6, its twist by $-68$.
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.