Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+624x+7148\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+624xz^2+7148z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+809325x+331080750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 477950 \) | = | $2 \cdot 5^{2} \cdot 11^{2} \cdot 79$ |
|
| Discriminant: | $\Delta$ | = | $-37339843750$ | = | $-1 \cdot 2 \cdot 5^{9} \cdot 11^{2} \cdot 79 $ |
|
| j-invariant: | $j$ | = | \( \frac{14245319}{19750} \) | = | $2^{-1} \cdot 5^{-3} \cdot 11 \cdot 79^{-1} \cdot 109^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.71427884694519057622090768510$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.49008932140492136842312924451$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8954547760939792$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.3902388019184047$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.78034101492190671557971288077$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.5606820298438134311594257615 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.560682030 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.780341 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 1.560682030\end{aligned}$$
Modular invariants
Modular form 477950.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 435456 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $11$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $79$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3160 = 2^{3} \cdot 5 \cdot 79 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 3159 & 2 \\ 3158 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 3159 & 0 \end{array}\right),\left(\begin{array}{rr} 1897 & 2 \\ 1897 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 161 & 2 \\ 161 & 3 \end{array}\right),\left(\begin{array}{rr} 791 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1581 & 2 \\ 1581 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[3160])$ is a degree-$14174532403200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3160\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 238975 = 5^{2} \cdot 11^{2} \cdot 79 \) |
| $5$ | additive | $18$ | \( 19118 = 2 \cdot 11^{2} \cdot 79 \) |
| $11$ | additive | $32$ | \( 3950 = 2 \cdot 5^{2} \cdot 79 \) |
| $79$ | nonsplit multiplicative | $80$ | \( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 477950h consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 95590q1, its twist by $5$.
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 $0$.