Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-1716666x-865861870\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-1716666xz^2-865861870z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2224799163x-40390977009258\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 31790 \) | = | $2 \cdot 5 \cdot 11 \cdot 17^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-38873796250190$ | = | $-1 \cdot 2 \cdot 5 \cdot 11^{5} \cdot 17^{6} $ |
|
| j-invariant: | $j$ | = | \( -\frac{23178622194826561}{1610510} \) | = | $-1 \cdot 2^{-1} \cdot 5^{-1} \cdot 11^{-5} \cdot 285121^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0616822527909295303130545994$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.64507558076282149018828729046$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0129363433875729$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.274599906388421$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.065947464758779271473225530443$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.6486866189694817868306382611 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $25$ = $5^2$ (exact) |
|
BSD formula
$$\begin{aligned} 1.648686619 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.065947 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 1.648686619\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 440000 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $17$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7480 = 2^{3} \cdot 5 \cdot 11 \cdot 17 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 439 & 0 \\ 0 & 7479 \end{array}\right),\left(\begin{array}{rr} 3401 & 2210 \\ 6885 & 3571 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 7425 & 7361 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 3741 & 2210 \\ 1105 & 3571 \end{array}\right),\left(\begin{array}{rr} 1871 & 2210 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6937 & 5950 \\ 5015 & 4251 \end{array}\right),\left(\begin{array}{rr} 7471 & 10 \\ 7470 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[7480])$ is a degree-$15882780672000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7480\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 15895 = 5 \cdot 11 \cdot 17^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 578 = 2 \cdot 17^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 2890 = 2 \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 110 = 2 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 31790o
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 110a2, its twist by $17$.
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 |
|---|---|---|---|
| $3$ | 3.1.440.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.0.36125.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.85184000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.2.1386579101562500000000.1 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | 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 | split | ord | nonsplit | ord | nonsplit | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 2 | 4 | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 1 | 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 $0$.