Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-93795x-12107593\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-93795xz^2-12107593z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-1500723x-776386674\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 11466 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $-10763393803185114$ | = | $-1 \cdot 2 \cdot 3^{6} \cdot 7^{6} \cdot 13^{7} $ |
|
| j-invariant: | $j$ | = | \( -\frac{1064019559329}{125497034} \) | = | $-1 \cdot 2^{-1} \cdot 3^{3} \cdot 13^{-7} \cdot 41^{3} \cdot 83^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8120557449650960634463060602$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.28979452610338456519600707002$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0626891964834324$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.936878540743459$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.13550574848825042688932289322$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 7 $ = $ 1\cdot1\cdot1\cdot7 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.94854023941775298822526025256 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 0.948540239 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.135506 \cdot 1.000000 \cdot 7}{1^2} \\ & \approx 0.948540239\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 74088 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
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 |
|---|---|---|
| $7$ | 7B.6.3 | 7.24.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 2171 & 14 \\ 2170 & 15 \end{array}\right),\left(\begin{array}{rr} 1576 & 735 \\ 441 & 1450 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 727 & 0 \\ 0 & 2183 \end{array}\right),\left(\begin{array}{rr} 1271 & 1656 \\ 378 & 167 \end{array}\right),\left(\begin{array}{rr} 736 & 735 \\ 357 & 1450 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 736 & 735 \\ 903 & 1450 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$40577531904$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 | $4$ | \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 1274 = 2 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 18 = 2 \cdot 3^{2} \) |
| $13$ | split multiplicative | $14$ | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 11466.n
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 26.b1, its twist by $21$.
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 |
|---|---|---|---|
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.64827.1 | \(\Z/7\Z\) | not in database |
| $8$ | 8.2.2399575035312.8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $14$ | 14.2.42528428123116916736.1 | \(\Z/7\Z\) | not in database |
| $18$ | 18.0.344721053590134031632826368.1 | \(\Z/14\Z\) | 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 | nonsplit | add | ord | add | ord | split | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 1 | - | 0 | - | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 |
| $\mu$-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
All $p$-adic regulators are identically $1$ since the rank is $0$.