Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-6584008x-51414671552\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-6584008xz^2-51414671552z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-8532875043x-2398674922807842\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(4304, -2152\right) \)
Integral points
\( \left(4304, -2152\right) \)
Invariants
Conductor: | \( 439530 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1123590185991000000000000 $ | = | $-1 \cdot 2^{12} \cdot 3^{3} \cdot 5^{12} \cdot 7^{7} \cdot 13^{3} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{268291321601301081721}{9550359000000000000} \) | = | $-1 \cdot 2^{-12} \cdot 3^{-3} \cdot 5^{-12} \cdot 7^{-1} \cdot 11^{3} \cdot 13^{-3} \cdot 23^{-1} \cdot 457^{3} \cdot 1283^{3}$ | 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: | $3.2951328406189098854060669131\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.3221777660912532328533905414\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9782064771169592\dots$ | |||
Szpiro ratio: | $4.837003505910458\dots$ |
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: | $0.037921734262153907965195796116\dots$ | 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: | $ 8 $ = $ 2\cdot1\cdot2\cdot2\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: | $2$ | 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) $ ≈ $ 1.2134954963889250548862654757 $ | 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
$\displaystyle 1.213495496 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{16 \cdot 0.037922 \cdot 1.000000 \cdot 8}{2^2} \approx 1.213495496$
Modular invariants
Modular form 439530.2.a.i
For more coefficients, see the Downloads section to the right.
Modular degree: | 79626240 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 6 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$3$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$5$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$7$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$13$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$23$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$2$ | 2B | 4.6.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 251160 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 23 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 251137 & 24 \\ 251136 & 25 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 249854 & 241931 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 71744 & 251139 \\ 72045 & 374 \end{array}\right),\left(\begin{array}{rr} 52329 & 10468 \\ 240740 & 198869 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 150697 & 24 \\ 50244 & 289 \end{array}\right),\left(\begin{array}{rr} 16 & 125583 \\ 208841 & 167354 \end{array}\right),\left(\begin{array}{rr} 10936 & 3 \\ 141501 & 251074 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 154576 & 21 \\ 134955 & 250786 \end{array}\right),\left(\begin{array}{rr} 188377 & 24 \\ 188274 & 250831 \end{array}\right)$.
The torsion field $K:=\Q(E[251160])$ is a degree-$1300922165247344640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/251160\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 439530.i
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 62790.v7, its twist by $-7$.
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 $0$.