Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-117x+4050\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-117xz^2+4050z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-9504x+2923965\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-18, 0)$ | $0$ | $2$ |
Integral points
\( \left(-18, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 10164 \) | = | $2^{2} \cdot 3 \cdot 7 \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-6846429744$ | = | $-1 \cdot 2^{4} \cdot 3^{8} \cdot 7^{2} \cdot 11^{3} $ |
|
| j-invariant: | $j$ | = | \( -\frac{8388608}{321489} \) | = | $-1 \cdot 2^{23} \cdot 3^{-8} \cdot 7^{-2}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.56743197479566251829807607130$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.26309090359057855418982053034$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.2365498124436336$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.2641142306213933$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $1.1069697894477673260202896282$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.2139395788955346520405792564 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.213939579 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.106970 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 2.213939579\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5760 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $2$ | $III$ | additive | 1 | 2 | 3 | 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 |
|---|---|---|
| $2$ | 2B | 8.12.0.31 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \), index $96$, genus $3$, and generators
$\left(\begin{array}{rr} 2465 & 16 \\ 1240 & 129 \end{array}\right),\left(\begin{array}{rr} 2773 & 16 \\ 8 & 129 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 3632 & 3603 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 533 & 26 \\ 2006 & 3145 \end{array}\right),\left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 1688 & 5 \\ 1059 & 32 \end{array}\right),\left(\begin{array}{rr} 2315 & 26 \\ 2666 & 3145 \end{array}\right),\left(\begin{array}{rr} 3681 & 16 \\ 3680 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 28 \\ 3620 & 2987 \end{array}\right)$.
The torsion field $K:=\Q(E[3696])$ is a degree-$326998425600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3696\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3388 = 2^{2} \cdot 7 \cdot 11^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $42$ | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 10164.k
consists of 2 curves linked by isogenies of
degree 2.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.85184.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.7256313856.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.17422409568256.6 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 7 | 11 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | nonsplit | add |
| $\lambda$-invariant(s) | - | 0 | 0 | - |
| $\mu$-invariant(s) | - | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.