Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+2694x+142820\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+2694xz^2+142820z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+3491397x+6652935702\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4, 390)$ | $0$ | $3$ |
Integral points
\( \left(4, 390\right) \), \( \left(4, -394\right) \)
Invariants
| Conductor: | $N$ | = | \( 8330 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $-10035365580800$ | = | $-1 \cdot 2^{12} \cdot 5^{2} \cdot 7^{8} \cdot 17 $ |
|
| j-invariant: | $j$ | = | \( \frac{375078431}{1740800} \) | = | $2^{-12} \cdot 5^{-2} \cdot 7 \cdot 13^{3} \cdot 17^{-1} \cdot 29^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1771405169624011391366686711$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.12013291574114106426689982453$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0000826672438765$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.1271880333238125$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.51968096365549127936416390552$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ ( 2^{2} \cdot 3 )\cdot2\cdot3\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
|
| Special value: | $ L(E,1)$ | ≈ | $4.1574477092439302349133112441 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 4.157447709 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.519681 \cdot 1.000000 \cdot 72}{3^2} \\ & \approx 4.157447709\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 16128 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 204 = 2^{2} \cdot 3 \cdot 17 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 199 & 6 \\ 198 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 201 & 202 \\ 194 & 197 \end{array}\right),\left(\begin{array}{rr} 103 & 6 \\ 105 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 37 & 6 \\ 111 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 18 & 193 \\ 85 & 171 \end{array}\right)$.
The torsion field $K:=\Q(E[204])$ is a degree-$22560768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/204\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 833 = 7^{2} \cdot 17 \) |
| $3$ | good | $2$ | \( 4165 = 5 \cdot 7^{2} \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1666 = 2 \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $26$ | \( 170 = 2 \cdot 5 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 8330.x
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 8330.r2, its twist by $-7$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.3332.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.754951232.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.3384009916875.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.10612255099320000.6 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.45872527737523318535404113095907000000000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | 17 |
|---|---|---|---|---|---|
| Reduction type | split | ord | nonsplit | add | nonsplit |
| $\lambda$-invariant(s) | 3 | 2 | 0 | - | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
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 $0$.