Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-3\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-3z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-3x-194\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4, 5)$ | $0.58097875960685951707331339128$ | $\infty$ |
Integral points
\( \left(4, 5\right) \), \( \left(4, -9\right) \)
Invariants
| Conductor: | $N$ | = | \( 567 \) | = | $3^{4} \cdot 7$ |
|
| Discriminant: | $\Delta$ | = | $-3969$ | = | $-1 \cdot 3^{4} \cdot 7^{2} $ |
|
| j-invariant: | $j$ | = | \( -\frac{9}{49} \) | = | $-1 \cdot 3^{2} \cdot 7^{-2}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.63055176279450505458022262458$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.99675585901720828504530437022$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1006981514469614$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.482681964058504$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.58097875960685951707331339128$ |
|
| Real period: | $\Omega$ | ≈ | $2.0002970970383927452104106433$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.3242602525651346455576345700 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.324260253 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.000297 \cdot 0.580979 \cdot 2}{1^2} \\ & \approx 2.324260253\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 56.4.0-4.a.1.1, level \( 56 = 2^{3} \cdot 7 \), index $4$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 29 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 17 & 4 \\ 34 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 51 & 49 \end{array}\right),\left(\begin{array}{rr} 29 & 25 \\ 2 & 37 \end{array}\right),\left(\begin{array}{rr} 53 & 4 \\ 52 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[56])$ is a degree-$774144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/56\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 81 = 3^{4} \) |
| $3$ | additive | $8$ | \( 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 81 = 3^{4} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 567a consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 567b1, its twist by $-3$.
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 |
|---|---|---|---|
| $3$ | 3.1.324.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.419904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.425329947.1 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | ord | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.