Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+y=x^3-61\)
|
(homogenize, simplify) |
|
\(y^2z+yz^2=x^3-61z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-3888\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 243 \) | = | $3^{5}$ |
|
| Discriminant: | $\Delta$ | = | $-1594323$ | = | $-1 \cdot 3^{13} $ |
|
| j-invariant: | $j$ | = | \( 0 \) | = | $0$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.13095411570425241597835352252$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3211174284280379149898691959$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9571157042857488$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $1.2249188952151323028374328392$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.2249188952151323028374328392 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.224918895 \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 1.224919 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 1.224918895\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 27 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There is only one prime $p$ 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 | 5 | 13 | 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 |
|---|---|---|
| $3$ | 3B.1.2 | 27.1944.55.49 |
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 |
|---|---|---|---|
| $3$ | additive | $4$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 243.b
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 243.b2, its twist by $-3$.
The minimal sextic twist of this elliptic curve is 27.a4, its sextic 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-6561.1-CMd1 |
| $3$ | 3.1.972.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.243.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.2834352.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.177147.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.1162261467.1 | \(\Z/9\Z\) | not in database |
| $9$ | 9.1.24794911296.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.98716277881700352.11 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/21\Z\) | not in database |
| $18$ | 18.0.4052555153018976267.1 | \(\Z/9\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.1844362878529525198848.1 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 |
|---|---|---|
| Reduction type | ss | add |
| $\lambda$-invariant(s) | 0,3 | - |
| $\mu$-invariant(s) | 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 $0$.