Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2+2128592x+2015205312\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z+2128592xz^2+2015205312z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+172415925x+1469601920250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(89608/9, 27136000/27)$ | $1.7546589853826827104855095420$ | $\infty$ |
| $(5448/49, 16281600/343)$ | $2.6074088121754587927618567568$ | $\infty$ |
Integral points
\((1362,\pm 86250)\)
Invariants
| Conductor: | $N$ | = | \( 402800 \) | = | $2^{4} \cdot 5^{2} \cdot 19 \cdot 53$ |
|
| Discriminant: | $\Delta$ | = | $-2372854507110400000000$ | = | $-1 \cdot 2^{31} \cdot 5^{8} \cdot 19 \cdot 53^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{16665594512227991}{37075851673600} \) | = | $2^{-19} \cdot 5^{-2} \cdot 11^{6} \cdot 19^{-1} \cdot 53^{-3} \cdot 2111^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7856812518858700758795767884$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2878151151088745791619650003$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9780438607124865$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.367811078904404$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.5235683711675866124517430634$ |
|
| Real period: | $\Omega$ | ≈ | $0.10094473788652755193921499183$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $21.918260329886323566697196620 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 21.918260330 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.100945 \cdot 4.523568 \cdot 48}{1^2} \\ & \approx 21.918260330\end{aligned}$$
Modular invariants
Modular form 402800.2.a.bg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15759360 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{23}^{*}$ | additive | -1 | 4 | 31 | 19 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $53$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 \( 8056 = 2^{3} \cdot 19 \cdot 53 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 4029 & 2 \\ 4029 & 3 \end{array}\right),\left(\begin{array}{rr} 6785 & 2 \\ 6785 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 8055 & 0 \end{array}\right),\left(\begin{array}{rr} 2281 & 2 \\ 2281 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8055 & 2 \\ 8054 & 3 \end{array}\right),\left(\begin{array}{rr} 2015 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[8056])$ is a degree-$731755749703680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8056\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 25175 = 5^{2} \cdot 19 \cdot 53 \) |
| $3$ | good | $2$ | \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 16112 = 2^{4} \cdot 19 \cdot 53 \) |
| $19$ | split multiplicative | $20$ | \( 21200 = 2^{4} \cdot 5^{2} \cdot 53 \) |
| $53$ | split multiplicative | $54$ | \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 402800bg consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 10070b1, its twist by $-20$.
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.