Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-5990639055x+178464471262629\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-5990639055xz^2+178464471262629z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-7763868215955x+8326554829252454574\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{219130352532872654}{4909844977489}, -\frac{22803457284577736023599}{10879317968484743513}\right)\) |
$\hat{h}(P)$ | ≈ | $33.975497961697286143781028902$ |
Torsion generators
\( \left(44686, -22343\right) \)
Integral points
\( \left(44686, -22343\right) \)
Invariants
Conductor: | \( 405042 \) | = | $2 \cdot 3 \cdot 11 \cdot 17 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $38373289533960741888 $ | = | $2^{10} \cdot 3 \cdot 11 \cdot 17^{6} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{505384091400037554067434625}{815656731648} \) | = | $2^{-10} \cdot 3^{-1} \cdot 5^{3} \cdot 11^{-1} \cdot 17^{-6} \cdot 31^{6} \cdot 97^{3} \cdot 1709^{3}$ | 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: | $3.9131643742422945268143428260\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.4409448846590742968098291101\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.1764898982289331\dots$ | |||
Szpiro ratio: | $6.130386591541099\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $33.975497961697286143781028902\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.093024498201658953761809064246\dots$ | 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\cdot1\cdot2\cdot2 $ | 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'(E,1) $ ≈ $ 6.3211072980767532879598743435 $ | 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
$\displaystyle 6.321107298 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.093024 \cdot 33.975498 \cdot 8}{2^2} \approx 6.321107298$
Modular invariants
Modular form 405042.2.a.s
For more coefficients, see the Downloads section to the right.
Modular degree: | 209018880 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{10}$ | Non-split multiplicative | 1 | 1 | 10 | 10 |
$3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$11$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$17$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$19$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
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 | 8.6.0.4 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5016 = 2^{3} \cdot 3 \cdot 11 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2639 & 0 \\ 0 & 5015 \end{array}\right),\left(\begin{array}{rr} 2870 & 2907 \\ 589 & 1312 \end{array}\right),\left(\begin{array}{rr} 1863 & 2698 \\ 494 & 533 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2509 & 1596 \\ 798 & 4561 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 4966 & 5007 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4618 & 2907 \\ 3477 & 1312 \end{array}\right),\left(\begin{array}{rr} 5005 & 12 \\ 5004 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[5016])$ is a degree-$1248141312000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5016\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 405042s
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1122m3, its twist by $-19$.
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.