Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-1755266x-765489279\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-1755266xz^2-765489279z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-28084251x-49019398090\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-979, 4215\right)\) | \(\left(-795, 11667\right)\) |
$\hat{h}(P)$ | ≈ | $2.1886015509206008043096348715$ | $2.4008303503761945281363232703$ |
Torsion generators
\( \left(-991, 495\right) \), \( \left(1505, -753\right) \)
Integral points
\( \left(-991, 495\right) \), \( \left(-979, 4215\right) \), \( \left(-979, -3237\right) \), \( \left(-795, 11667\right) \), \( \left(-795, -10873\right) \), \( \left(-523, 3303\right) \), \( \left(-523, -2781\right) \), \( \left(1505, -753\right) \), \( \left(1557, 15783\right) \), \( \left(1557, -17341\right) \), \( \left(2909, 135435\right) \), \( \left(2909, -138345\right) \), \( \left(5093, 347283\right) \), \( \left(5093, -352377\right) \), \( \left(8417, 757839\right) \), \( \left(8417, -766257\right) \), \( \left(24609, 3842415\right) \), \( \left(24609, -3867025\right) \), \( \left(98381, 30805815\right) \), \( \left(98381, -30904197\right) \)
Invariants
Conductor: | \( 489762 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $92674544140192944384 $ | = | $2^{8} \cdot 3^{10} \cdot 7^{4} \cdot 13^{6} \cdot 23^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{169967019783457}{26337394944} \) | = | $2^{-8} \cdot 3^{-4} \cdot 7^{-4} \cdot 13^{3} \cdot 23^{-2} \cdot 4261^{3}$ | 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: | $2.5546081955447156157321257998\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.72282737247989240200775946055\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.96307678039626\dots$ | |||
Szpiro ratio: | $4.178702042863392\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $5.0063946080433896036750628360\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.13253139473177656265219506464\dots$ | 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: | $ 512 $ = $ 2^{3}\cdot2^{2}\cdot2\cdot2^{2}\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: | $4$ | 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! $ ≈ $ 21.232142719412360774868436694 $ | 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
$\displaystyle 21.232142719 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.132531 \cdot 5.006395 \cdot 512}{4^2} \approx 21.232142719$
Modular invariants
Modular form 489762.2.a.dg
For more coefficients, see the Downloads section to the right.
Modular degree: | 18874368 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 4 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$7$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$13$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$23$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
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 |
---|---|---|
$2$ | 2Cs | 8.24.0.10 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 50232 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 23 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 33487 & 0 \\ 0 & 50231 \end{array}\right),\left(\begin{array}{rr} 31591 & 42510 \\ 9906 & 7723 \end{array}\right),\left(\begin{array}{rr} 19969 & 29952 \\ 45396 & 35413 \end{array}\right),\left(\begin{array}{rr} 15455 & 0 \\ 0 & 50231 \end{array}\right),\left(\begin{array}{rr} 50225 & 8 \\ 50224 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 50228 & 50229 \end{array}\right),\left(\begin{array}{rr} 14353 & 39936 \\ 2028 & 9049 \end{array}\right),\left(\begin{array}{rr} 47659 & 7410 \\ 5148 & 45397 \end{array}\right)$.
The torsion field $K:=\Q(E[50232])$ is a degree-$5420509021863936$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/50232\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 489762.dg
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 966.g4, 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.