Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-91922x-10110380\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-91922xz^2-10110380z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-1470755x-648535074\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-140, 70)$ | $0$ | $2$ |
Integral points
\( \left(-140, 70\right) \)
Invariants
| Conductor: | $N$ | = | \( 489566 \) | = | $2 \cdot 7 \cdot 11^{2} \cdot 17^{2}$ |
|
| Discriminant: | $\Delta$ | = | $5349768638052608$ | = | $2^{8} \cdot 7^{4} \cdot 11^{6} \cdot 17^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{9869198625}{614656} \) | = | $2^{-8} \cdot 3^{3} \cdot 5^{3} \cdot 7^{-4} \cdot 11^{3} \cdot 13^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7694116152481302721793533849$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.13783935716510901991400205855$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0498004125880718$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.503452732113282$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.27525511907055542999833261032$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.1010204762822217199933304413 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.101020476 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.275255 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 1.101020476\end{aligned}$$
Modular invariants
Modular form 489566.2.a.x
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2949120 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $2$ | $III$ | 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 | 16.48.1.35 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1904 = 2^{4} \cdot 7 \cdot 17 \), index $192$, genus $9$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 8 & 1557 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 64 & 205 \end{array}\right),\left(\begin{array}{rr} 1361 & 4 \\ 1100 & 49 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 1840 & 1811 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 8 & 65 \end{array}\right),\left(\begin{array}{rr} 1889 & 16 \\ 1888 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 956 \\ 12 & 287 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1236 & 5 \\ 819 & 44 \end{array}\right)$.
The torsion field $K:=\Q(E[1904])$ is a degree-$20214448128$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1904\Z)$.
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 | $4$ | \( 2057 = 11^{2} \cdot 17 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 69938 = 2 \cdot 11^{2} \cdot 17^{2} \) |
| $11$ | additive | $62$ | \( 4046 = 2 \cdot 7 \cdot 17^{2} \) |
| $17$ | additive | $82$ | \( 1694 = 2 \cdot 7 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 489566.x
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 4046.o1, its twist by $-11$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.