Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-119430x-491724\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-119430xz^2-491724z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1910883x-33381218\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-68, 2738\right)\) | \(\left(-159, 3882\right)\) |
$\hat{h}(P)$ | ≈ | $0.98404564364111557238977617648$ | $2.0820359286890351333162285054$ |
Torsion generators
\( \left(348, -174\right) \)
Integral points
\( \left(-237, 3921\right) \), \( \left(-237, -3684\right) \), \( \left(-159, 3882\right) \), \( \left(-159, -3723\right) \), \( \left(-132, 3666\right) \), \( \left(-132, -3534\right) \), \( \left(-68, 2738\right) \), \( \left(-68, -2670\right) \), \( \left(-15, 1146\right) \), \( \left(-15, -1131\right) \), \( \left(348, -174\right) \), \( \left(517, 8445\right) \), \( \left(517, -8962\right) \), \( \left(1788, 73266\right) \), \( \left(1788, -75054\right) \), \( \left(5340, 386706\right) \), \( \left(5340, -392046\right) \), \( \left(9903, 979896\right) \), \( \left(9903, -989799\right) \)
Invariants
Conductor: | \( 471510 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 31$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $108906526900454400 $ | = | $2^{10} \cdot 3^{7} \cdot 5^{2} \cdot 13^{7} \cdot 31 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{53540005609}{30950400} \) | = | $2^{-10} \cdot 3^{-1} \cdot 5^{-2} \cdot 13^{-1} \cdot 31^{-1} \cdot 3769^{3}$ | 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: | $1.9585969698574375903520018198\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.12681614679261437662763548056\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9297516717491048\dots$ | |||
Szpiro ratio: | $3.573649166568941\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.9020375295302496423389499985\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.28089283485590839215903009486\dots$ | 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 $ = $ 2\cdot2^{2}\cdot2\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! $ ≈ $ 8.5482994187532863118470999864 $ | 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
$\displaystyle 8.548299419 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.280893 \cdot 1.902038 \cdot 64}{2^2} \approx 8.548299419$
Modular invariants
Modular form 471510.2.a.e
For more coefficients, see the Downloads section to the right.
Modular degree: | 6021120 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{10}$ | Non-split multiplicative | 1 | 1 | 10 | 10 |
$3$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$13$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
$31$ | $1$ | $I_{1}$ | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48360 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 31 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 18137 & 30226 \\ 30224 & 18135 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 29642 & 1 \\ 9359 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 48357 & 4 \\ 48356 & 5 \end{array}\right),\left(\begin{array}{rr} 3722 & 1 \\ 22319 & 0 \end{array}\right),\left(\begin{array}{rr} 24181 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 32242 & 1 \\ 32239 & 0 \end{array}\right),\left(\begin{array}{rr} 29017 & 4 \\ 9674 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[48360])$ is a degree-$69004991397888000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48360\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 471510e
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 12090l1, 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.