Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2+2984091689x+407564856839509\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z+2984091689xz^2+407564856839509z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+3867382828269x+19015287949961704494\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-60898, 30449)$ | $0$ | $2$ |
Integral points
\( \left(-60898, 30449\right) \)
Invariants
| Conductor: | $N$ | = | \( 459186 \) | = | $2 \cdot 3 \cdot 7 \cdot 13 \cdot 29^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-73459354728835623682281673064448$ | = | $-1 \cdot 2^{48} \cdot 3^{2} \cdot 7^{3} \cdot 13^{2} \cdot 29^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{4940514904764290195189663}{123497771750673547788288} \) | = | $2^{-48} \cdot 3^{-2} \cdot 7^{-3} \cdot 13^{-2} \cdot 29^{-2} \cdot 47^{3} \cdot 337^{3} \cdot 10753^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.7946982631743963685148688769$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.1110503481811593549232328607$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0866977221990364$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.198072891636664$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.014568489289725634853230650748$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot1\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.46619165727122031530338082394 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
|
BSD formula
$$\begin{aligned} 0.466191657 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.014568 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 0.466191657\end{aligned}$$
Modular invariants
Modular form 459186.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1548288000 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{48}$ | nonsplit multiplicative | 1 | 1 | 48 | 48 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $29$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 | 16.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9744 = 2^{4} \cdot 3 \cdot 7 \cdot 29 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3665 & 16 \\ 1418 & 7599 \end{array}\right),\left(\begin{array}{rr} 9729 & 16 \\ 9728 & 17 \end{array}\right),\left(\begin{array}{rr} 4192 & 5 \\ 8307 & 9730 \end{array}\right),\left(\begin{array}{rr} 6509 & 16 \\ 2992 & 9429 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 9740 & 9741 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 9646 & 9731 \end{array}\right),\left(\begin{array}{rr} 6035 & 9728 \\ 9328 & 315 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 2444 & 2565 \end{array}\right)$.
The torsion field $K:=\Q(E[9744])$ is a degree-$8448450232320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9744\Z)$.
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 | $4$ | \( 5887 = 7 \cdot 29^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 10933 = 13 \cdot 29^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 65598 = 2 \cdot 3 \cdot 13 \cdot 29^{2} \) |
| $13$ | split multiplicative | $14$ | \( 35322 = 2 \cdot 3 \cdot 7 \cdot 29^{2} \) |
| $29$ | additive | $450$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 459186e
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 15834q1, its twist by $29$.
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 $0$.