Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-10196215x-13750404403\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-10196215xz^2-13750404403z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-13214294667x-641340653398074\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3877, 68866)$ | $1.0410658242275526140810070139$ | $\infty$ |
Integral points
\( \left(3877, 68866\right) \), \( \left(3877, -72744\right) \), \( \left(53007, 12154846\right) \), \( \left(53007, -12207854\right) \)
Invariants
| Conductor: | $N$ | = | \( 424830 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-13787606904560311200000$ | = | $-1 \cdot 2^{8} \cdot 3 \cdot 5^{5} \cdot 7^{7} \cdot 17^{8} $ |
|
| j-invariant: | $j$ | = | \( -\frac{142843688929}{16800000} \) | = | $-1 \cdot 2^{-8} \cdot 3^{-1} \cdot 5^{-5} \cdot 7^{-1} \cdot 17 \cdot 19^{3} \cdot 107^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9840588468028848130024821381$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.12229487623775077361678268780$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9003718974946426$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.646124782985865$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0410658242275526140810070139$ |
|
| Real period: | $\Omega$ | ≈ | $0.041966304546175261946870973384$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 240 $ = $ 2^{3}\cdot1\cdot5\cdot2\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $10.485524503715624983184856888 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 10.485524504 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.041966 \cdot 1.041066 \cdot 240}{1^2} \\ & \approx 10.485524504\end{aligned}$$
Modular invariants
Modular form 424830.2.a.fu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 37601280 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $17$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 419 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 2 \\ 337 & 3 \end{array}\right),\left(\begin{array}{rr} 211 & 2 \\ 211 & 3 \end{array}\right),\left(\begin{array}{rr} 241 & 2 \\ 241 & 3 \end{array}\right),\left(\begin{array}{rr} 419 & 2 \\ 418 & 3 \end{array}\right),\left(\begin{array}{rr} 281 & 2 \\ 281 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[420])$ is a degree-$2229534720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/420\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 212415 = 3 \cdot 5 \cdot 7^{2} \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 141610 = 2 \cdot 5 \cdot 7^{2} \cdot 17^{2} \) |
| $5$ | split multiplicative | $6$ | \( 84966 = 2 \cdot 3 \cdot 7^{2} \cdot 17^{2} \) |
| $7$ | additive | $32$ | \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 424830.fu consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 60690.bq1, its twist by $-119$.
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.