Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-5665x-164950\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-5665xz^2-164950z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-7341867x-7585770906\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-189/4, 185/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 11055 \) | = | $3 \cdot 5 \cdot 11 \cdot 67$ |
|
| Discriminant: | $\Delta$ | = | $215430675405$ | = | $3 \cdot 5 \cdot 11^{8} \cdot 67 $ |
|
| j-invariant: | $j$ | = | \( \frac{20106118884162961}{215430675405} \) | = | $3^{-1} \cdot 5^{-1} \cdot 11^{-8} \cdot 13^{6} \cdot 67^{-1} \cdot 1609^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.99035348991990328153387487841$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.99035348991990328153387487841$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9697034959062909$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.031925654246443$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.55065589092300532461771867974$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.1013117818460106492354373595 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
|
BSD formula
$$\begin{aligned} 1.101311782 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.550656 \cdot 1.000000 \cdot 2}{2^2} \\ & \approx 1.101311782\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12800 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $67$ | $1$ | $I_{1}$ | 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 | 8.12.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7043 & 7036 \\ 7082 & 3021 \end{array}\right),\left(\begin{array}{rr} 3019 & 3018 \\ 1018 & 5035 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 5368 & 3 \\ 5365 & 2 \end{array}\right),\left(\begin{array}{rr} 8033 & 8 \\ 8032 & 9 \end{array}\right),\left(\begin{array}{rr} 1088 & 3 \\ 1685 & 2 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 8034 & 8035 \end{array}\right),\left(\begin{array}{rr} 4828 & 1 \\ 1631 & 6 \end{array}\right)$.
The torsion field $K:=\Q(E[8040])$ is a degree-$14632011694080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8040\Z)$.
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 | $2$ | \( 1005 = 3 \cdot 5 \cdot 67 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3685 = 5 \cdot 11 \cdot 67 \) |
| $5$ | split multiplicative | $6$ | \( 2211 = 3 \cdot 11 \cdot 67 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 1005 = 3 \cdot 5 \cdot 67 \) |
| $67$ | nonsplit multiplicative | $68$ | \( 165 = 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 11055c
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{1005}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-201}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-201})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.6432000.2 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 11 | 67 |
|---|---|---|---|---|---|
| Reduction type | ord | nonsplit | split | nonsplit | nonsplit |
| $\lambda$-invariant(s) | 4 | 0 | 1 | 0 | 0 |
| $\mu$-invariant(s) | 2 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.