Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2-146279x-1461915\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z-146279xz^2-1461915z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-189578259x-65363435730\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-378, 189)$ | $0$ | $2$ |
| $(-10, 5)$ | $0$ | $2$ |
Integral points
\( \left(-378, 189\right) \), \( \left(-10, 5\right) \)
Invariants
| Conductor: | $N$ | = | \( 422142 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$ |
|
| Discriminant: | $\Delta$ | = | $199474994683527696$ | = | $2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2} \cdot 23^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{2338337977417}{1347477264} \) | = | $2^{-4} \cdot 3^{-2} \cdot 7^{-2} \cdot 13^{3} \cdot 19^{-2} \cdot 23^{-2} \cdot 1021^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0091017024172982831902660505$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.44135459445272343778688963460$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9241246685961948$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6511288933793846$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.26592339727218816798785521779$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2\cdot2\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.0636935890887526719514208712 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.063693589 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.265923 \cdot 1.000000 \cdot 64}{4^2} \\ & \approx 1.063693589\end{aligned}$$
Modular invariants
Modular form 422142.2.a.o
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4866048 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $23$ | $4$ | $I_{2}^{*}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 36708 = 2^{2} \cdot 3 \cdot 7 \cdot 19 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 24473 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 36705 & 4 \\ 36704 & 5 \end{array}\right),\left(\begin{array}{rr} 31919 & 36706 \\ 0 & 36707 \end{array}\right),\left(\begin{array}{rr} 18357 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 31467 & 2 \\ 15730 & 36707 \end{array}\right),\left(\begin{array}{rr} 17391 & 2 \\ 25114 & 36707 \end{array}\right)$.
The torsion field $K:=\Q(E[36708])$ is a degree-$6366119799029760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/36708\Z)$.
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 | $4$ | \( 529 = 23^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 140714 = 2 \cdot 7 \cdot 19 \cdot 23^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 60306 = 2 \cdot 3 \cdot 19 \cdot 23^{2} \) |
| $19$ | nonsplit multiplicative | $20$ | \( 22218 = 2 \cdot 3 \cdot 7 \cdot 23^{2} \) |
| $23$ | additive | $288$ | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 422142.o
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 18354.c2, its twist by $-23$.
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 $0$.