Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-136363612x+612908803936\)
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(homogenize, simplify) |
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\(y^2z=x^3-136363612xz^2+612908803936z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-136363612x+612908803936\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(8470, 256036\right)\)
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| $\hat{h}(P)$ | ≈ | $0.48692591467551306150670163451$ |
Integral points
\((6776,\pm 4840)\), \((8470,\pm 256036)\)
Invariants
| Conductor: | \( 445280 \) | = | $2^{5} \cdot 5 \cdot 11^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-1228521546095022080 $ | = | $-1 \cdot 2^{12} \cdot 5 \cdot 11^{8} \cdot 23^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{319387394710714176}{1399205} \) | = | $-1 \cdot 2^{6} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{3} \cdot 11 \cdot 23^{-4} \cdot 3659^{3}$ | 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: | $3.0995739879790099128858899375\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $0.80782995888681757409402876407\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.990378361798107\dots$ | |||
| Szpiro ratio: | $5.213264557826692\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $0.48692591467551306150670163451\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.18389897236077952314768472810\dots$ | 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^{2}\cdot1\cdot3\cdot2^{2} $ | 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) $ ≈ $ 4.2981684155836543146497772058 $ | 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
$\displaystyle 4.298168416 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.183899 \cdot 0.486926 \cdot 48}{1^2} \approx 4.298168416$
Modular invariants
Modular form 445280.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 57987072 | 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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | -1 | 5 | 12 | 0 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $23$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 label 20.2.0.a.1, level \( 20 = 2^{2} \cdot 5 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 19 & 0 \end{array}\right),\left(\begin{array}{rr} 17 & 2 \\ 17 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 11 & 3 \end{array}\right),\left(\begin{array}{rr} 19 & 2 \\ 18 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[20])$ is a degree-$23040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 605 = 5 \cdot 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 89056 = 2^{5} \cdot 11^{2} \cdot 23 \) |
| $11$ | additive | $52$ | \( 3680 = 2^{5} \cdot 5 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 19360 = 2^{5} \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 445280d consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 445280e1, 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.