Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+5121936x-8659529852\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+5121936xz^2-8659529852z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+81950973x-554127959554\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(3312, 209594\right)\)
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| $\hat{h}(P)$ | ≈ | $2.3740296425136742031652269552$ |
Torsion generators
\( \left(\frac{5123}{4}, -\frac{5123}{8}\right) \)
Integral points
\( \left(3312, 209594\right) \), \( \left(3312, -212906\right) \), \( \left(6224, 510986\right) \), \( \left(6224, -517210\right) \)
Invariants
| Conductor: | \( 441090 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 29$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-40984683415027157812500 $ | = | $-1 \cdot 2^{2} \cdot 3^{8} \cdot 5^{8} \cdot 13^{10} \cdot 29 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{4223169036960119}{11647532812500} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-8} \cdot 13^{-4} \cdot 29^{-1} \cdot 161639^{3}$ | 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: | $3.0225211849495534400515450539\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.1907403618847302263271787147\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.9390084776053046\dots$ | |||
| Szpiro ratio: | $4.561566752688005\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $2.3740296425136742031652269552\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.058910178299575093192507602174\dots$ | 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^{3}\cdot2^{2}\cdot1 $ | 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: | $2$ | 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) $ ≈ $ 8.9506886098532522329779929916 $ | 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
$\displaystyle 8.950688610 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.058910 \cdot 2.374030 \cdot 256}{2^2} \approx 8.950688610$
Modular invariants
Modular form 441090.2.a.bw
For more coefficients, see the Downloads section to the right.
| Modular degree: | 66060288 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
| $5$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
| $13$ | $4$ | $I_{4}^{*}$ | Additive | 1 | 2 | 10 | 4 |
| $29$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9048 = 2^{3} \cdot 3 \cdot 13 \cdot 29 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 9042 & 9043 \end{array}\right),\left(\begin{array}{rr} 7655 & 6024 \\ 6492 & 5999 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5659 & 5658 \\ 4914 & 3403 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 9041 & 8 \\ 9040 & 9 \end{array}\right),\left(\begin{array}{rr} 6031 & 0 \\ 0 & 9047 \end{array}\right),\left(\begin{array}{rr} 5656 & 4155 \\ 2643 & 8704 \end{array}\right),\left(\begin{array}{rr} 6976 & 3 \\ 213 & 6034 \end{array}\right)$.
The torsion field $K:=\Q(E[9048])$ is a degree-$27457463255040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9048\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 441090.bw
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 11310.c4, its twist by $-39$.
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.