Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-4512910x+2651349572\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-4512910xz^2+2651349572z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-5848731387x+123718911825366\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(182, 42758\right)\) | \(\left(1838, 22886\right)\) |
$\hat{h}(P)$ | ≈ | $0.52479935499950445759040257893$ | $2.0418648324268228558573405625$ |
Torsion generators
\( \left(\frac{2591}{4}, -\frac{2591}{8}\right) \)
Integral points
\( \left(-2086, 55718\right) \), \( \left(-2086, -53632\right) \), \( \left(-1888, 67598\right) \), \( \left(-1888, -65710\right) \), \( \left(-1336, 80018\right) \), \( \left(-1336, -78682\right) \), \( \left(-196, 59498\right) \), \( \left(-196, -59302\right) \), \( \left(182, 42758\right) \), \( \left(182, -42940\right) \), \( \left(554, 17648\right) \), \( \left(554, -18202\right) \), \( \left(1838, 22886\right) \), \( \left(1838, -24724\right) \), \( \left(2288, 64466\right) \), \( \left(2288, -66754\right) \), \( \left(7634, 638918\right) \), \( \left(7634, -646552\right) \), \( \left(73184, 19753298\right) \), \( \left(73184, -19826482\right) \)
Invariants
Conductor: | \( 491970 \) | = | $2 \cdot 3 \cdot 5 \cdot 23^{2} \cdot 31$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $2844665970694114161600 $ | = | $2^{6} \cdot 3^{18} \cdot 5^{2} \cdot 23^{6} \cdot 31 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{68663623745397169}{19216056254400} \) | = | $2^{-6} \cdot 3^{-18} \cdot 5^{-2} \cdot 31^{-1} \cdot 43^{3} \cdot 89^{3} \cdot 107^{3}$ | 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: | $2.8237595973600023099482712314\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $1.2560124893954274645448948155\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0695661960289142\dots$ | |||
Szpiro ratio: | $4.3934229447552315\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.0125174533614943795545466605\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.13334926110132330390543393016\dots$ | 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);
|
Tamagawa product: | $ 864 $ = $ ( 2 \cdot 3 )\cdot( 2 \cdot 3^{2} )\cdot2\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])
|
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! $ ≈ $ 29.163986119716952681243085847 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 29.163986120 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.133349 \cdot 1.012517 \cdot 864}{2^2} \approx 29.163986120$
Modular invariants
Modular form 491970.2.a.co
For more coefficients, see the Downloads section to the right.
Modular degree: | 36495360 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$3$ | $18$ | $I_{18}$ | Split multiplicative | -1 | 1 | 18 | 18 |
$5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$23$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$31$ | $1$ | $I_{1}$ | 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 | 2.3.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 42780 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \cdot 31 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 18010 & 1863 \\ 5313 & 37192 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 42769 & 12 \\ 42768 & 13 \end{array}\right),\left(\begin{array}{rr} 19229 & 29762 \\ 2484 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 16739 & 0 \\ 0 & 42779 \end{array}\right),\left(\begin{array}{rr} 8557 & 7452 \\ 12282 & 1933 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17826 & 11017 \\ 17825 & 3566 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 42730 & 42771 \end{array}\right)$.
The torsion field $K:=\Q(E[42780])$ is a degree-$5495675682816000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/42780\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 491970co
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 930n2, its twist by $-23$.
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.