Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+322373x-2058811158\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+322373xz^2-2058811158z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+322373x-2058811158\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 40432 \) | = | $2^{4} \cdot 7 \cdot 19^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-1833264015930066599936$ | = | $-1 \cdot 2^{17} \cdot 7^{7} \cdot 19^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{53261199}{26353376} \) | = | $2^{-5} \cdot 3^{3} \cdot 7^{-7} \cdot 19 \cdot 47^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7591108543945230387692087108$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.10300435439028408934595830142$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1541630705988888$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.318368106883493$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.069475970425914549277039928994$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.1116155268146327884326388639 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
|
BSD formula
$$\begin{aligned} 1.111615527 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.069476 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 1.111615527\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1149120 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | -1 | 4 | 17 | 5 |
| $7$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $19$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1064 = 2^{3} \cdot 7 \cdot 19 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1051 & 14 \\ 1050 & 15 \end{array}\right),\left(\begin{array}{rr} 267 & 546 \\ 805 & 745 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 533 & 14 \\ 539 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 275 & 1050 \\ 294 & 39 \end{array}\right),\left(\begin{array}{rr} 265 & 1050 \\ 0 & 1063 \end{array}\right)$.
The torsion field $K:=\Q(E[1064])$ is a degree-$3971358720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1064\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 2527 = 7 \cdot 19^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $146$ | \( 112 = 2^{4} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 40432m
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 5054a1, its twist by $76$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.20216.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.22886452736.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.8340544.1 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.0.279596236589374562737309351936.1 | \(\Z/14\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ss | ord | nonsplit | ord | ord | ss | add | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 0,0 | 2 | 0 | 0 | 0 | 0,0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
| $\mu$-invariant(s) | - | 0,0 | 0 | 0 | 0 | 0 | 0,0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.