Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2+7850718x-17999735436\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z+7850718xz^2-17999735436z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+10174529853x-839948274453186\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{13169012}{1369}, \frac{49068488478}{50653}\right)\) |
$\hat{h}(P)$ | ≈ | $11.169721379201363225122288177$ |
Torsion generators
\( \left(1684, -842\right) \)
Integral points
\( \left(1684, -842\right) \)
Invariants
Conductor: | \( 479370 \) | = | $2 \cdot 3 \cdot 5 \cdot 19 \cdot 29^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-170982395701110374400000 $ | = | $-1 \cdot 2^{20} \cdot 3^{5} \cdot 5^{5} \cdot 19^{2} \cdot 29^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{89962967236397039}{287450726400000} \) | = | $2^{-20} \cdot 3^{-5} \cdot 5^{-5} \cdot 19^{-2} \cdot 29^{3} \cdot 15451^{3}$ | 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: | $3.1411844895082521533928939845\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.4575365745150151398012579683\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0281324592856305\dots$ | |||
Szpiro ratio: | $4.644472150327595\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $11.169721379201363225122288177\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.051995275023140911953684079553\dots$ | 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: | $ 40 $ = $ 2\cdot1\cdot5\cdot2\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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 5.8077273504343170024840173224 $ | 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
$\displaystyle 5.807727350 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.051995 \cdot 11.169721 \cdot 40}{2^2} \approx 5.807727350$
Modular invariants
Modular form 479370.2.a.l
For more coefficients, see the Downloads section to the right.
Modular degree: | 60480000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{20}$ | Non-split multiplicative | 1 | 1 | 20 | 20 |
$3$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
$5$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$19$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$29$ | $2$ | $I_0^{*}$ | 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 | 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 \( 33060 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \cdot 29 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 12181 & 1160 \\ 6670 & 11601 \end{array}\right),\left(\begin{array}{rr} 18239 & 0 \\ 0 & 33059 \end{array}\right),\left(\begin{array}{rr} 20156 & 25085 \\ 5655 & 23926 \end{array}\right),\left(\begin{array}{rr} 21896 & 25085 \\ 19575 & 25202 \end{array}\right),\left(\begin{array}{rr} 33041 & 20 \\ 33040 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 16531 & 1160 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 32820 & 32711 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[33060])$ is a degree-$644948656128000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/33060\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 479370l
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 570l1, its twist by $29$.
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.