Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+1703013x+1896235094\) | (homogenize, simplify) |
\(y^2z=x^3+1703013xz^2+1896235094z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+1703013x+1896235094\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-130, 40898\right)\) |
$\hat{h}(P)$ | ≈ | $5.0234205231780822846976995640$ |
Torsion generators
\( \left(-806, 0\right) \)
Integral points
\( \left(-806, 0\right) \), \((-130,\pm 40898)\)
Invariants
Conductor: | \( 425880 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1869452472006748661760 $ | = | $-1 \cdot 2^{11} \cdot 3^{8} \cdot 5 \cdot 7^{8} \cdot 13^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{75798394558}{259416045} \) | = | $2 \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{-8} \cdot 3359^{3}$ | 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: | $2.7646958968434992585456026927\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.29753015826539284452210690879\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9884224063998598\dots$ | |||
Szpiro ratio: | $4.339592346210173\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $5.0234205231780822846976995640\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.10504406030458988748996583352\dots$ | 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: | $ 64 $ = $ 1\cdot2\cdot1\cdot2^{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: | $2$ | 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) $ ≈ $ 8.4428878139525273306931139611 $ | 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
$\displaystyle 8.442887814 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.105044 \cdot 5.023421 \cdot 64}{2^2} \approx 8.442887814$
Modular invariants
Modular form 425880.2.a.eb
For more coefficients, see the Downloads section to the right.
Modular degree: | 18874368 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II^{*}$ | Additive | -1 | 3 | 11 | 0 |
$3$ | $2$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$7$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$13$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21840 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 21742 & 21827 \end{array}\right),\left(\begin{array}{rr} 13987 & 1664 \\ 9776 & 5355 \end{array}\right),\left(\begin{array}{rr} 21825 & 16 \\ 21824 & 17 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 21839 \end{array}\right),\left(\begin{array}{rr} 18721 & 20176 \\ 6968 & 8529 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 21836 & 21837 \end{array}\right),\left(\begin{array}{rr} 21646 & 17225 \\ 16965 & 19306 \end{array}\right),\left(\begin{array}{rr} 7736 & 6721 \\ 19903 & 16810 \end{array}\right),\left(\begin{array}{rr} 18916 & 12181 \\ 9399 & 5890 \end{array}\right)$.
The torsion field $K:=\Q(E[21840])$ is a degree-$155817722511360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21840\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 425880.eb
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 840.d6, 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.