Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-82305009x+2630267735013\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-82305009xz^2+2630267735013z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1316880147x+168335818160686\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(\frac{19383}{4}, \frac{12231867}{8}\right)\) | \(\left(3127, 1548749\right)\) |
$\hat{h}(P)$ | ≈ | $1.6299478710478885495607388389$ | $2.4366792188963904548777984984$ |
Torsion generators
\( \left(-\frac{63117}{4}, \frac{63117}{8}\right) \)
Integral points
\( \left(-15573, 375324\right) \), \( \left(-15573, -359751\right) \), \( \left(-7851, 1674996\right) \), \( \left(-7851, -1667145\right) \), \( \left(3127, 1548749\right) \), \( \left(3127, -1551876\right) \), \( \left(13977, 2044899\right) \), \( \left(13977, -2058876\right) \), \( \left(19377, 2873124\right) \), \( \left(19377, -2892501\right) \), \( \left(3206877, 5741169999\right) \), \( \left(3206877, -5744376876\right) \)
Invariants
Conductor: | \( 402930 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-2952979662393951416015625000 $ | = | $-1 \cdot 2^{3} \cdot 3^{10} \cdot 5^{20} \cdot 11^{6} \cdot 37 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{47744008200656797609}{2286529541015625000} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-4} \cdot 5^{-20} \cdot 37^{-1} \cdot 3627769^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $3.9510451194652188662598694178\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.2027913387319787485312750104\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0586106987102104\dots$ | |||
Szpiro ratio: | $5.479351030383004\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.4790426646103460513031148590\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.037425617207232648847894601960\dots$ | 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: | $ 320 $ = $ 1\cdot2^{2}\cdot( 2^{2} \cdot 5 )\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! $ ≈ $ 10.416425521066999391645769867 $ | 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
$\displaystyle 10.416425521 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.037426 \cdot 3.479043 \cdot 320}{2^2} \approx 10.416425521$
Modular invariants
Modular form 402930.2.a.bk
For more coefficients, see the Downloads section to the right.
Modular degree: | 235929600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$5$ | $20$ | $I_{20}$ | Split multiplicative | -1 | 1 | 20 | 20 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$37$ | $1$ | $I_{1}$ | 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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48840 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 48834 & 48835 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 40888 & 47553 \\ 25707 & 2014 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 42736 & 10923 \\ 14619 & 13168 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 16279 & 0 \\ 0 & 48839 \end{array}\right),\left(\begin{array}{rr} 19537 & 41448 \\ 1188 & 19273 \end{array}\right),\left(\begin{array}{rr} 13319 & 0 \\ 0 & 48839 \end{array}\right),\left(\begin{array}{rr} 43528 & 39963 \\ 21285 & 29602 \end{array}\right),\left(\begin{array}{rr} 48833 & 8 \\ 48832 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[48840])$ is a degree-$17733591760896000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48840\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 402930bk
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1110f4, 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.