Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+40180x+9045008\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+40180xz^2+9045008z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+40180x+9045008\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-14, 2912)$ | $0.22938089030132743528186299514$ | $\infty$ |
Integral points
\((-14,\pm 2912)\), \((77,\pm 3549)\), \((4978,\pm 351520)\)
Invariants
| Conductor: | $N$ | = | \( 40768 \) | = | $2^{6} \cdot 7^{2} \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $-39494402524315648$ | = | $-1 \cdot 2^{18} \cdot 7^{4} \cdot 13^{7} $ |
|
| j-invariant: | $j$ | = | \( \frac{11397810375}{62748517} \) | = | $3^{3} \cdot 5^{3} \cdot 7^{2} \cdot 13^{-7} \cdot 41^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8666319505840480257662497267$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.17827446339235895993861729670$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0540820890244733$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.2916430840924145$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.22938089030132743528186299514$ |
|
| Real period: | $\Omega$ | ≈ | $0.26229049880623984858325550600$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 84 $ = $ 2^{2}\cdot3\cdot7 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $5.0538119632353828153484838107 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.053811963 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.262290 \cdot 0.229381 \cdot 84}{1^2} \\ & \approx 5.053811963\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 215040 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | -1 | 6 | 18 | 0 |
| $7$ | $3$ | $IV$ | additive | 1 | 2 | 4 | 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.6.2 | 7.24.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 728 = 2^{3} \cdot 7 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 608 & 721 \\ 651 & 370 \end{array}\right),\left(\begin{array}{rr} 363 & 0 \\ 0 & 727 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 187 & 14 \\ 350 & 247 \end{array}\right),\left(\begin{array}{rr} 377 & 714 \\ 378 & 713 \end{array}\right),\left(\begin{array}{rr} 720 & 721 \\ 553 & 370 \end{array}\right),\left(\begin{array}{rr} 715 & 14 \\ 714 & 15 \end{array}\right)$.
The torsion field $K:=\Q(E[728])$ is a degree-$845365248$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/728\Z)$.
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 | $2$ | \( 637 = 7^{2} \cdot 13 \) |
| $7$ | additive | $20$ | \( 64 = 2^{6} \) |
| $13$ | split multiplicative | $14$ | \( 3136 = 2^{6} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 40768.cc
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 637.c2, its twist by $-8$.
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 |
|---|---|---|---|
| $3$ | 3.1.2548.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.337599808.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.8605184.1 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.6.196843269980502900896750895104.1 | \(\Z/14\Z\) | 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 | add | ss | ss | add | ord | split | ord | ord | ord | ord | ss | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | 1,1 | 1,1 | - | 1 | 2 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0,0 | 0,0 | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.