Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+y=x^3-x^2+4269180x-10910193631\)
|
(homogenize, simplify) |
|
\(y^2z+yz^2=x^3-x^2z+4269180xz^2-10910193631z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+5532856848x-508959599752368\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(52089/16, 12390619/64)$ | $1.9870529676413790269925925102$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 458913 \) | = | $3 \cdot 7 \cdot 13 \cdot 41^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-56388366667482790293171$ | = | $-1 \cdot 3^{8} \cdot 7^{7} \cdot 13^{3} \cdot 41^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{1811564780171264}{11870974573731} \) | = | $2^{12} \cdot 3^{-8} \cdot 7^{-7} \cdot 13^{-3} \cdot 19^{3} \cdot 401^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0472967437482520718474427856$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1905107103960981699140610991$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0472072211574832$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.583153461695148$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9870529676413790269925925102$ |
|
| Real period: | $\Omega$ | ≈ | $0.055690945009971064133178987352$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 168 $ = $ 2\cdot7\cdot3\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $18.591024068873063084385999453 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 18.591024069 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.055691 \cdot 1.987053 \cdot 168}{1^2} \\ & \approx 18.591024069\end{aligned}$$
Modular invariants
Modular form 458913.2.a.j
For more coefficients, see the Downloads section to the right.
| Modular degree: | 47308800 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $13$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $41$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 182 = 2 \cdot 7 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 181 & 2 \\ 180 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 181 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 15 & 3 \end{array}\right),\left(\begin{array}{rr} 157 & 2 \\ 157 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[182])$ is a degree-$158505984$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/182\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 152971 = 7 \cdot 13 \cdot 41^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 11767 = 7 \cdot 41^{2} \) |
| $7$ | split multiplicative | $8$ | \( 65559 = 3 \cdot 13 \cdot 41^{2} \) |
| $13$ | split multiplicative | $14$ | \( 35301 = 3 \cdot 7 \cdot 41^{2} \) |
| $41$ | additive | $842$ | \( 273 = 3 \cdot 7 \cdot 13 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 458913j consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 273b1, its twist by $41$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.