Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-826275x+920281250\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-826275xz^2+920281250z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-826275x+920281250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-1250, 0\right) \)
Integral points
\( \left(-1250, 0\right) \)
Invariants
| Conductor: | \( 471600 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 131$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $-329764519540224000000 $ | = | $-1 \cdot 2^{15} \cdot 3^{7} \cdot 5^{6} \cdot 131^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( -\frac{1337180541913}{7067998104} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 23^{3} \cdot 131^{-4} \cdot 479^{3}$ | 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: | $2.6212627912856417885584544833\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $0.57409051017459144614322007677\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $1.166204397364557\dots$ | |||
| Szpiro ratio: | $4.195039931865506\dots$ | |||
BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $0.14833709823837420170932691983\dots$ | 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: | $ 64 $ = $ 2\cdot2\cdot2^{2}\cdot2^{2} $ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
| Special value: | $ L(E,1) $ ≈ $ 2.3733935718139872273492307173 $ | 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
$\displaystyle 2.373393572 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.148337 \cdot 1.000000 \cdot 64}{2^2} \approx 2.373393572$
Modular invariants
Modular form 471600.2.a.de
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14155776 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{7}^{*}$ | Additive | -1 | 4 | 15 | 3 |
| $3$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
| $5$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
| $131$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15720 = 2^{3} \cdot 3 \cdot 5 \cdot 131 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 15714 & 15715 \end{array}\right),\left(\begin{array}{rr} 3143 & 0 \\ 0 & 15719 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 11524 & 12575 \\ 1025 & 15714 \end{array}\right),\left(\begin{array}{rr} 15713 & 8 \\ 15712 & 9 \end{array}\right),\left(\begin{array}{rr} 389 & 6680 \\ 8230 & 9819 \end{array}\right),\left(\begin{array}{rr} 5896 & 7475 \\ 14545 & 8286 \end{array}\right),\left(\begin{array}{rr} 14281 & 9440 \\ 6820 & 6321 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[15720])$ is a degree-$215458873344000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15720\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 471600de
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 786e4, its twist by $60$.
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 $0$.