Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-43411905x-107747532099\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-43411905xz^2-107747532099z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-694590483x-6896536644818\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-3342, 1671\right) \)
Integral points
\( \left(-3342, 1671\right) \)
Invariants
Conductor: | \( 422730 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 61$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $219755223716490000000000 $ | = | $2^{10} \cdot 3^{11} \cdot 5^{10} \cdot 7^{5} \cdot 11^{2} \cdot 61 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{12411455694361059171563281}{301447494810000000000} \) | = | $2^{-10} \cdot 3^{-5} \cdot 5^{-10} \cdot 7^{-5} \cdot 11^{-2} \cdot 61^{-1} \cdot 231530161^{3}$ | 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: | $3.2631770920506989991881326211\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.7138709477166441534905100026\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9632519983313432\dots$ | |||
Szpiro ratio: | $4.969114264034543\dots$ |
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: | $0.058903376323990117664589942125\dots$ | 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: | $ 160 $ = $ 2\cdot2^{2}\cdot2\cdot5\cdot2\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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 2.3561350529596047065835976850 $ | 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
$\displaystyle 2.356135053 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.058903 \cdot 1.000000 \cdot 160}{2^2} \approx 2.356135053$
Modular invariants
Modular form 422730.2.a.r
For more coefficients, see the Downloads section to the right.
Modular degree: | 71680000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{10}$ | Non-split multiplicative | 1 | 1 | 10 | 10 |
$3$ | $4$ | $I_{5}^{*}$ | Additive | -1 | 2 | 11 | 5 |
$5$ | $2$ | $I_{10}$ | Non-split multiplicative | 1 | 1 | 10 | 10 |
$7$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$11$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$61$ | $1$ | $I_{1}$ | 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 | 2.3.0.1 |
$5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 563640 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 61 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 140911 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 281821 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 112731 & 14 \\ 563620 & 563547 \end{array}\right),\left(\begin{array}{rr} 563634 & 563627 \\ 375845 & 184 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 563400 & 563291 \end{array}\right),\left(\begin{array}{rr} 161046 & 13 \\ 160955 & 563456 \end{array}\right),\left(\begin{array}{rr} 526686 & 13 \\ 147755 & 563456 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 563621 & 20 \\ 563620 & 21 \end{array}\right),\left(\begin{array}{rr} 307441 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[563640])$ is a degree-$44521489642291200000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/563640\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 422730.r
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 140910.dd3, its twist by $-3$.
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 $0$.