Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-10008691x+12886006205\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-10008691xz^2+12886006205z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-12971263563x+601248419291142\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{14705}{4}, \frac{14705}{8}\right) \)
Integral points
None
Invariants
| Conductor: | \( 491970 \) | = | $2 \cdot 3 \cdot 5 \cdot 23^{2} \cdot 31$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-7575508976862643202940 $ | = | $-1 \cdot 2^{2} \cdot 3 \cdot 5 \cdot 23^{6} \cdot 31^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{749011598724977281}{51173462246460} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-1} \cdot 5^{-1} \cdot 31^{-8} \cdot 71^{3} \cdot 12791^{3}$ | 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: | $2.9495727501794178798989497282\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.3818256422148430344955733123\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0019471514445586\dots$ | |||
| Szpiro ratio: | $4.584258916843539\dots$ | |||
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: | $0.12971684260095593826877696948\dots$ | 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: | $ 16 $ = $ 2\cdot1\cdot1\cdot2^{2}\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: | $2$ | 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: | $16$ = $4^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L(E,1) $ ≈ $ 8.3018779264611800492017260464 $ | 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
$\displaystyle 8.301877926 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{16 \cdot 0.129717 \cdot 1.000000 \cdot 16}{2^2} \approx 8.301877926$
Modular invariants
Modular form 491970.2.a.ci
For more coefficients, see the Downloads section to the right.
| Modular degree: | 46137344 | 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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $23$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
| $31$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
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 | 8.24.0.90 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 171120 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \cdot 31 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1496 & 14881 \\ 11983 & 133930 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 171116 & 171117 \end{array}\right),\left(\begin{array}{rr} 47128 & 14881 \\ 34799 & 133930 \end{array}\right),\left(\begin{array}{rr} 59519 & 0 \\ 0 & 171119 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 171105 & 16 \\ 171104 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 171022 & 171107 \end{array}\right),\left(\begin{array}{rr} 71623 & 66976 \\ 14214 & 158425 \end{array}\right),\left(\begin{array}{rr} 7453 & 66976 \\ 83444 & 46185 \end{array}\right),\left(\begin{array}{rr} 5521 & 66976 \\ 163208 & 22449 \end{array}\right)$.
The torsion field $K:=\Q(E[171120])$ is a degree-$703446487400448000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/171120\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 491970ci
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 930o6, its twist by $-23$.
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 $0$.