Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-1222708x-520495054\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-1222708xz^2-520495054z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1584628947x-24279463340946\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-638, 323\right)\) | \(\left(2386, 99539\right)\) |
$\hat{h}(P)$ | ≈ | $2.3191121848752239358302098388$ | $4.1517876434959037296183511134$ |
Torsion generators
\( \left(-639, 319\right) \)
Integral points
\( \left(-639, 319\right) \), \( \left(-638, 323\right) \), \( \left(-638, 314\right) \), \( \left(1297, 8063\right) \), \( \left(1297, -9361\right) \), \( \left(2386, 99539\right) \), \( \left(2386, -101926\right) \), \( \left(3457, 189503\right) \), \( \left(3457, -192961\right) \), \( \left(6417, 502639\right) \), \( \left(6417, -509057\right) \), \( \left(45586, 9707354\right) \), \( \left(45586, -9752941\right) \)
Invariants
Conductor: | \( 482790 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $219859507042560 $ | = | $2^{8} \cdot 3^{6} \cdot 5 \cdot 7 \cdot 11^{6} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{114113060120923921}{124104960} \) | = | $2^{-8} \cdot 3^{-6} \cdot 5^{-1} \cdot 7^{-1} \cdot 19^{-1} \cdot 485041^{3}$ | 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: | $2.0394070448395577984901960726\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.84045940844037252645922428362\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9574479119060725\dots$ | |||
Szpiro ratio: | $4.100401595571707\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $9.3762726493890700032789697022\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.14357173118598075043736411208\dots$ | 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: | $ 48 $ = $ 2\cdot( 2 \cdot 3 )\cdot1\cdot1\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^{(2)}(E,1)/2! $ ≈ $ 16.154012356134613157875785587 $ | 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
$\displaystyle 16.154012356 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.143572 \cdot 9.376273 \cdot 48}{2^2} \approx 16.154012356$
Modular invariants
Modular form 482790.2.a.dr
For more coefficients, see the Downloads section to the right.
Modular degree: | 9830400 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 4 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
$3$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$19$ | $1$ | $I_{1}$ | 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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 351120 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 314656 & 255365 \\ 310035 & 287266 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 351022 & 351107 \end{array}\right),\left(\begin{array}{rr} 291281 & 255376 \\ 76010 & 327471 \end{array}\right),\left(\begin{array}{rr} 191519 & 0 \\ 0 & 351119 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 255376 \\ 87780 & 87781 \end{array}\right),\left(\begin{array}{rr} 65528 & 191521 \\ 124399 & 319210 \end{array}\right),\left(\begin{array}{rr} 344752 & 255365 \\ 159555 & 287266 \end{array}\right),\left(\begin{array}{rr} 308573 & 255376 \\ 10384 & 223125 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 351116 & 351117 \end{array}\right),\left(\begin{array}{rr} 351105 & 16 \\ 351104 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[351120])$ is a degree-$9662411078369280000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/351120\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 482790.dr
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 3990.bb4, its twist by $-11$.
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.