Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-23888078x+6090614648048\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-23888078xz^2+6090614648048z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-30958948467x+284163809896184526\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{74793}{4}, \frac{74789}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-149586:74789:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-673134, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 490110 \) | = | $2 \cdot 3 \cdot 5 \cdot 17 \cdot 31^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-16024431553321480805202462000$ | = | $-1 \cdot 2^{4} \cdot 3^{20} \cdot 5^{3} \cdot 17^{4} \cdot 31^{7} $ |
|
| j-invariant: | $j$ | = | \( -\frac{1698623579042432281}{18055622637267102000} \) | = | $-1 \cdot 2^{-4} \cdot 3^{-20} \cdot 5^{-3} \cdot 17^{-4} \cdot 31^{-1} \cdot 1193161^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.0909027622063927208528509874$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.3739091599638195978882688251$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0451364403778787$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.525611229332001$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.031356492698511117411395869254$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 480 $ = $ 2\cdot( 2^{2} \cdot 5 )\cdot3\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $3.7627791238213340893675043105 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 3.762779124 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.031356 \cdot 1.000000 \cdot 480}{2^2} \\ & \approx 3.762779124\end{aligned}$$
Modular invariants
Modular form 490110.2.a.bi
For more coefficients, see the Downloads section to the right.
| Modular degree: | 383385600 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $17$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $31$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 63240 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \cdot 31 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 63233 & 8 \\ 63232 & 9 \end{array}\right),\left(\begin{array}{rr} 23723 & 23716 \\ 23762 & 55341 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 26512 & 63237 \\ 24475 & 63238 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 12656 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 55339 & 55338 \\ 39538 & 7915 \end{array}\right),\left(\begin{array}{rr} 42161 & 8 \\ 42164 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 63234 & 63235 \end{array}\right),\left(\begin{array}{rr} 18601 & 8 \\ 11164 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[63240])$ is a degree-$51564169396224000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/63240\Z)$.
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 | $4$ | \( 4805 = 5 \cdot 31^{2} \) |
| $3$ | split multiplicative | $4$ | \( 32674 = 2 \cdot 17 \cdot 31^{2} \) |
| $5$ | split multiplicative | $6$ | \( 32674 = 2 \cdot 17 \cdot 31^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 28830 = 2 \cdot 3 \cdot 5 \cdot 31^{2} \) |
| $31$ | additive | $512$ | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 490110bi
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 15810c4, its twist by $-31$.
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 $0$.