Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-671400x+219858624\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-671400xz^2+219858624z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-10742403x+14060209534\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 466578 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-1480298766111897984$ | = | $-1 \cdot 2^{7} \cdot 3^{13} \cdot 7^{2} \cdot 23^{6} $ |
|
| j-invariant: | $j$ | = | \( -\frac{6329617441}{279936} \) | = | $-1 \cdot 2^{-7} \cdot 3^{-7} \cdot 7 \cdot 967^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2515298157852836619637127406$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.18984179468923157998817841767$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.032335220248665$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.979000265207337$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.26630131946681204621603068673$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2^{2}\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.1304105557344963697282454938 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.130410556 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.266301 \cdot 1.000000 \cdot 8}{1^2} \\ & \approx 2.130410556\end{aligned}$$
Modular invariants
Modular form 466578.2.a.bj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8279040 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $3$ | $4$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $23$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $7$ | 7B.6.1 | 7.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 967 & 3542 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3851 & 14 \\ 3850 & 15 \end{array}\right),\left(\begin{array}{rr} 2899 & 1610 \\ 0 & 1795 \end{array}\right),\left(\begin{array}{rr} 1448 & 161 \\ 1127 & 3702 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 1007 & 0 \\ 0 & 3863 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 3704 & 3703 \\ 2093 & 162 \end{array}\right)$.
The torsion field $K:=\Q(E[3864])$ is a degree-$413653008384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3864\Z)$.
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 | $4$ | \( 233289 = 3^{2} \cdot 7^{2} \cdot 23^{2} \) |
| $3$ | additive | $8$ | \( 51842 = 2 \cdot 7^{2} \cdot 23^{2} \) |
| $7$ | additive | $14$ | \( 4761 = 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 466578.bj
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 294.f1, its twist by $69$.
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 $0$.