Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-36x+82\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-36xz^2+82z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-46035x+3975534\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5, 3)$ | $0$ | $6$ |
Integral points
\( \left(-7, 3\right) \), \( \left(2, 3\right) \), \( \left(2, -6\right) \), \( \left(5, 3\right) \), \( \left(5, -9\right) \)
Invariants
| Conductor: | $N$ | = | \( 138 \) | = | $2 \cdot 3 \cdot 23$ |
|
| Discriminant: | $\Delta$ | = | $-268272$ | = | $-1 \cdot 2^{4} \cdot 3^{6} \cdot 23 $ |
|
| j-invariant: | $j$ | = | \( -\frac{4956477625}{268272} \) | = | $-1 \cdot 2^{-4} \cdot 3^{-6} \cdot 5^{3} \cdot 11^{3} \cdot 23^{-1} \cdot 31^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.19986975544740434581771591190$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.19986975544740434581771591190$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.950715771479093$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.548856749091305$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $3.0602034221727587957503010239$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot( 2 \cdot 3 )\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.0200678073909195985834336746 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.020067807 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.060203 \cdot 1.000000 \cdot 12}{6^2} \\ & \approx 1.020067807\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 16 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $23$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 276 = 2^{2} \cdot 3 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 226 & 267 \end{array}\right),\left(\begin{array}{rr} 101 & 2 \\ 96 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 265 & 12 \\ 264 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 27 & 118 \\ 182 & 269 \end{array}\right),\left(\begin{array}{rr} 142 & 3 \\ 141 & 268 \end{array}\right)$.
The torsion field $K:=\Q(E[276])$ is a degree-$12824064$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/276\Z)$.
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 | $4$ | \( 23 \) |
| $3$ | split multiplicative | $4$ | \( 46 = 2 \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 6 = 2 \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 138b
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | 4.2.3312.2 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.120891312.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.21317168016.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.5802782976.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.64246599740592.4 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.26566813997047684912095297877393152.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 23 |
|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit |
| $\lambda$-invariant(s) | 1 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.