Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-4679994480x-92617453339200\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-4679994480xz^2-92617453339200z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-379079552907x-67519260722935494\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-22095, 0\right) \)
Integral points
\( \left(-22095, 0\right) \)
Invariants
Conductor: | \( 493680 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $2854365971544325785489349017600 $ | = | $2^{27} \cdot 3^{2} \cdot 5^{2} \cdot 11^{18} \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1562225332123379392365961}{393363080510106009600} \) | = | $2^{-15} \cdot 3^{-2} \cdot 5^{-2} \cdot 11^{-12} \cdot 17^{-1} \cdot 83^{3} \cdot 223^{3} \cdot 6269^{3}$ | 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: | $4.5537556779212456230038786042\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.6616608609621150415556746938\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0171250683975468\dots$ | |||
Szpiro ratio: | $5.981344606687099\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.018590908945269327050798381495\dots$ | 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: | $ 64 $ = $ 2^{2}\cdot2\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])
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Analytic order of Ш: | $9$ = $3^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 2.6770908881187830953149669353 $ | 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
$\displaystyle 2.677090888 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{9 \cdot 0.018591 \cdot 1.000000 \cdot 64}{2^2} \approx 2.677090888$
Modular invariants
Modular form 493680.2.a.dn
For more coefficients, see the Downloads section to the right.
Modular degree: | 796262400 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{19}^{*}$ | Additive | -1 | 4 | 27 | 15 |
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$11$ | $4$ | $I_{12}^{*}$ | Additive | -1 | 2 | 18 | 12 |
$17$ | $1$ | $I_{1}$ | 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 | 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 \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 8977 & 4092 \\ 11022 & 2113 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1605 & 22352 \\ 13882 & 4003 \end{array}\right),\left(\begin{array}{rr} 22429 & 12 \\ 22428 & 13 \end{array}\right),\left(\begin{array}{rr} 4079 & 0 \\ 0 & 22439 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 14290 & 12243 \\ 13233 & 8152 \end{array}\right),\left(\begin{array}{rr} 2410 & 12243 \\ 13893 & 8152 \end{array}\right),\left(\begin{array}{rr} 7481 & 4092 \\ 20746 & 2113 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 22390 & 22431 \end{array}\right)$.
The torsion field $K:=\Q(E[22440])$ is a degree-$381186736128000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 493680dn
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 5610r4, its twist by $44$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.