Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-20508411x+35747446966\) | (homogenize, simplify) |
\(y^2z=x^3-20508411xz^2+35747446966z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-20508411x+35747446966\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{21497}{4}, \frac{2273355}{8}\right)\) |
$\hat{h}(P)$ | ≈ | $6.6053672768223458826649380781$ |
Torsion generators
\( \left(2618, 0\right) \)
Integral points
\( \left(2618, 0\right) \)
Invariants
Conductor: | \( 426888 \) | = | $2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $2800556514844526592 $ | = | $2^{11} \cdot 3^{8} \cdot 7^{6} \cdot 11^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{3065617154}{9} \) | = | $2 \cdot 3^{-2} \cdot 1153^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
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: | $2.7690037497409848070394922238\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-0.58759002103319516354090800004\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.2105879594055395\dots$ | |||
Szpiro ratio: | $4.791832214729783\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $6.6053672768223458826649380781\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.22182846064559590633022772138\dots$ | 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: | $ 16 $ = $ 1\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)
|
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);
|
Special value: | $ L'(E,1) $ ≈ $ 5.8610338200651710178522179427 $ | 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
$\displaystyle 5.861033820 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.221828 \cdot 6.605367 \cdot 16}{2^2} \approx 5.861033820$
Modular invariants
Modular form 426888.2.a.bp
For more coefficients, see the Downloads section to the right.
Modular degree: | 15728640 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | not computed* (one of 4 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II^{*}$ | Additive | -1 | 3 | 11 | 0 |
$3$ | $2$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$7$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$11$ | $2$ | $I_0^{*}$ | 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 | 16.48.0.210 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 2111 & 0 \\ 0 & 3695 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 3692 & 3693 \end{array}\right),\left(\begin{array}{rr} 1156 & 3157 \\ 231 & 1618 \end{array}\right),\left(\begin{array}{rr} 307 & 1232 \\ 1232 & 3387 \end{array}\right),\left(\begin{array}{rr} 3681 & 16 \\ 3680 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 3023 & 0 \\ 0 & 3695 \end{array}\right),\left(\begin{array}{rr} 3235 & 2464 \\ 2310 & 2773 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 3598 & 3683 \end{array}\right)$.
The torsion field $K:=\Q(E[3696])$ is a degree-$163499212800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3696\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 426888bp
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 24a5, its twist by $-231$.
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.