Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2-24837676x-47655046064\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z-24837676xz^2-47655046064z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-32189628771x-2222910984733794\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(190744/25, 56403668/125)$ | $8.2335908793561466611604833290$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 422142 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-27538813257274368$ | = | $-1 \cdot 2^{10} \cdot 3^{2} \cdot 7^{7} \cdot 19^{3} \cdot 23^{2} $ |
|
| j-invariant: | $j$ | = | \( -\frac{3203324534532292752802873}{52058248123392} \) | = | $-1 \cdot 2^{-10} \cdot 3^{-2} \cdot 7^{-7} \cdot 19^{-3} \cdot 23 \cdot 29^{3} \cdot 97^{3} \cdot 18427^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6995448553347597523948209220$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1769624860132348039270287834$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0113261367140698$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.84032658270706$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.2335908793561466611604833290$ |
|
| Real period: | $\Omega$ | ≈ | $0.033813631617519419907104543466$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot2\cdot1\cdot3\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $3.3408913066069982922575650926 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.340891307 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.033814 \cdot 8.233591 \cdot 12}{1^2} \\ & \approx 3.340891307\end{aligned}$$
Modular invariants
Modular form 422142.2.a.b
For more coefficients, see the Downloads section to the right.
| Modular degree: | 19837440 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $19$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $23$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
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 \( 532 = 2^{2} \cdot 7 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 267 & 2 \\ 267 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 531 & 0 \end{array}\right),\left(\begin{array}{rr} 477 & 2 \\ 477 & 3 \end{array}\right),\left(\begin{array}{rr} 381 & 2 \\ 381 & 3 \end{array}\right),\left(\begin{array}{rr} 531 & 2 \\ 530 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[532])$ is a degree-$11914076160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/532\Z)$.
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 | $4$ | \( 70357 = 7 \cdot 19 \cdot 23^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 7406 = 2 \cdot 7 \cdot 23^{2} \) |
| $5$ | good | $2$ | \( 211071 = 3 \cdot 7 \cdot 19 \cdot 23^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 60306 = 2 \cdot 3 \cdot 19 \cdot 23^{2} \) |
| $19$ | split multiplicative | $20$ | \( 22218 = 2 \cdot 3 \cdot 7 \cdot 23^{2} \) |
| $23$ | additive | $112$ | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 422142.b consists of this curve only.
Twists
This elliptic curve is its own minimal quadratic twist.
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.