Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2+498559x-302552097\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z+498559xz^2-302552097z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+40383252x-220681628496\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(2251, 110592\right)\) |
$\hat{h}(P)$ | ≈ | $1.5197263360990890801175389204$ |
Integral points
\((538,\pm 11031)\), \((2251,\pm 110592)\)
Invariants
Conductor: | \( 404544 \) | = | $2^{6} \cdot 3 \cdot 7^{2} \cdot 43$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-47518786339171467264 $ | = | $-1 \cdot 2^{32} \cdot 3^{7} \cdot 7^{6} \cdot 43 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{444369620591}{1540767744} \) | = | $2^{-14} \cdot 3^{-7} \cdot 13^{3} \cdot 43^{-1} \cdot 587^{3}$ | 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: | $2.4586427684984573355741621930\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.44596692313088271889563763909\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9966384920336764\dots$ | |||
Szpiro ratio: | $4.0726313925296065\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.5197263360990890801175389204\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.10255776948432750157823636250\dots$ | 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: | $ 28 $ = $ 2^{2}\cdot7\cdot1\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: | $1$ | 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) $ ≈ $ 4.3640728111935359561698074562 $ | 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
$\displaystyle 4.364072811 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.102558 \cdot 1.519726 \cdot 28}{1^2} \approx 4.364072811$
Modular invariants
Modular form 404544.2.a.fg
For more coefficients, see the Downloads section to the right.
Modular degree: | 12192768 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{22}^{*}$ | Additive | 1 | 6 | 32 | 14 |
$3$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$7$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$43$ | $1$ | $I_{1}$ | Non-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 |
---|---|---|
$7$ | 7B.6.1 | 7.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7224 = 2^{3} \cdot 3 \cdot 7 \cdot 43 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 3611 & 0 \\ 0 & 7223 \end{array}\right),\left(\begin{array}{rr} 1807 & 14 \\ 0 & 1291 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 4276 & 7217 \\ 6643 & 3618 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 1799 & 3606 \end{array}\right),\left(\begin{array}{rr} 3625 & 7210 \\ 3626 & 7209 \end{array}\right),\left(\begin{array}{rr} 2416 & 7 \\ 2401 & 3606 \end{array}\right),\left(\begin{array}{rr} 7211 & 14 \\ 7210 & 15 \end{array}\right)$.
The torsion field $K:=\Q(E[7224])$ is a degree-$5167392620544$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7224\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 404544.fg
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 258.f2, its twist by $-56$.
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.