Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-62640x-8571744\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-62640xz^2-8571744z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1002243x-549593858\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
| $P$ | = |
\(\left(333, 2556\right)\)
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\(\left(\frac{10407}{4}, \frac{1045923}{8}\right)\)
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| $\hat{h}(P)$ | ≈ | $0.81859536310321071759660933734$ | $4.2073181563976825350990803080$ |
Integral points
\( \left(315, 1539\right) \), \( \left(315, -1854\right) \), \( \left(333, 2556\right) \), \( \left(333, -2889\right) \), \( \left(1785, 73704\right) \), \( \left(1785, -75489\right) \), \( \left(59623, 14528606\right) \), \( \left(59623, -14588229\right) \)
Invariants
| Conductor: | \( 468270 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 43$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-16126818822496800 $ | = | $-1 \cdot 2^{5} \cdot 3^{7} \cdot 5^{2} \cdot 11^{8} \cdot 43 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{173945761}{103200} \) | = | $-1 \cdot 2^{-5} \cdot 3^{-1} \cdot 5^{-2} \cdot 11 \cdot 43^{-1} \cdot 251^{3}$ | 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: | $1.8123620594279180229705824046\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-0.33554093343838385210166926584\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.9837145164610193\dots$ | |||
| Szpiro ratio: | $3.4813092383458715\dots$ | |||
BSD invariants
| Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $3.4293750504171179574045836092\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.14689426253429887115918985832\dots$ | 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: | $ 24 $ = $ 1\cdot2^{2}\cdot2\cdot3\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^{(2)}(E,1)/2! $ ≈ $ 12.090132455629117265741992243 $ | 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
$\displaystyle 12.090132456 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.146894 \cdot 3.429375 \cdot 24}{1^2} \approx 12.090132456$
Modular invariants
Modular form 468270.2.a.r
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3548160 | 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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
| $3$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
| $5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $3$ | $IV^{*}$ | Additive | -1 | 2 | 8 | 0 |
| $43$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1032 = 2^{3} \cdot 3 \cdot 43 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 689 & 2 \\ 689 & 3 \end{array}\right),\left(\begin{array}{rr} 775 & 2 \\ 775 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1031 & 0 \end{array}\right),\left(\begin{array}{rr} 433 & 2 \\ 433 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1031 & 2 \\ 1030 & 3 \end{array}\right),\left(\begin{array}{rr} 517 & 2 \\ 517 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[1032])$ is a degree-$123033157632$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1032\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 468270r consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 156090bf1, its twist by $33$.
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.