Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-420377x-104802288\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-420377xz^2-104802288z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-6726027x-6714072442\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{1134136259040}{3031713721}, \frac{31117903097059843}{166929189191981}\right) \) | $23.295780546219719324001732943$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-62446676559001440:31117903097059843:166929189191981]\) | $23.295780546219719324001732943$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{4539576749881}{3031713721}, -\frac{175764702759092}{166929189191981}\right) \) | $23.295780546219719324001732943$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 480249 \) | = | $3^{4} \cdot 7^{2} \cdot 11^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $7031325609$ | = | $3^{4} \cdot 7^{2} \cdot 11^{6} $ |
|
| j-invariant: | $j$ | = | \( 1168429123449 \) | = | $3^{2} \cdot 7 \cdot 2647^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6238099267300144115407162870$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.26566016406775964180622937153$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0481648553214244$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.857215333739148$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $23.295780546219719324001732943$ |
|
| Real period: | $\Omega$ | ≈ | $0.18749508982150710862570922086$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.3678444659755842079127314037 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.367844466 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.187495 \cdot 23.295781 \cdot 1}{1^2} \\ & \approx 4.367844466\end{aligned}$$
Modular invariants
Modular form 480249.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1499400 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $11$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 |
|---|---|---|---|
| $2$ | 2Cn | 2.2.0.1 | $2$ |
| $7$ | 7B | 7.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2772 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 11 \), index $576$, genus $16$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 28 & 1 \end{array}\right),\left(\begin{array}{rr} 386 & 2277 \\ 693 & 2663 \end{array}\right),\left(\begin{array}{rr} 1376 & 253 \\ 2387 & 23 \end{array}\right),\left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 14 \\ 2548 & 2563 \end{array}\right),\left(\begin{array}{rr} 1387 & 1540 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 251 & 0 \\ 0 & 2771 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 14 & 29 \end{array}\right),\left(\begin{array}{rr} 2745 & 28 \\ 2744 & 29 \end{array}\right)$.
The torsion field $K:=\Q(E[2772])$ is a degree-$17244057600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2772\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 5929 = 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $14$ | \( 9801 = 3^{4} \cdot 11^{2} \) |
| $11$ | additive | $62$ | \( 3969 = 3^{4} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 480249s
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 3969f2, its twist by $33$.
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.