Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2+7344062x-48658120219\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z+7344062xz^2-48658120219z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+9517904325x-2270336025494250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(11955/4, -11959/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 476850 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-1048287962623864746093750$ | = | $-1 \cdot 2 \cdot 3^{5} \cdot 5^{14} \cdot 11^{4} \cdot 17^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{116149984977671}{2779502343750} \) | = | $2^{-1} \cdot 3^{-5} \cdot 5^{-8} \cdot 11^{-4} \cdot 97^{3} \cdot 503^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2893093791825343396498385260$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0679837509373761122246915505$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.042494685403093$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.798443250852477$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.042377157526631445074280335380$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot1\cdot2^{2}\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.7121380817044124847539414643 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
|
BSD formula
$$\begin{aligned} 2.712138082 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.042377 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.712138082\end{aligned}$$
Modular invariants
Modular form 476850.2.a.gz
For more coefficients, see the Downloads section to the right.
| Modular degree: | 78643200 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $4$ | $I_{8}^{*}$ | additive | 1 | 2 | 14 | 8 |
| $11$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $17$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 22433 & 8 \\ 22432 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13463 & 0 \\ 0 & 22439 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 14876 & 17425 \\ 11815 & 19806 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 22434 & 22435 \end{array}\right),\left(\begin{array}{rr} 18361 & 4760 \\ 8500 & 19041 \end{array}\right),\left(\begin{array}{rr} 8416 & 1955 \\ 5185 & 21846 \end{array}\right),\left(\begin{array}{rr} 15811 & 3400 \\ 16490 & 5781 \end{array}\right),\left(\begin{array}{rr} 7919 & 0 \\ 0 & 22439 \end{array}\right)$.
The torsion field $K:=\Q(E[22440])$ is a degree-$762373472256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.
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 | $4$ | \( 21675 = 3 \cdot 5^{2} \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 158950 = 2 \cdot 5^{2} \cdot 11 \cdot 17^{2} \) |
| $5$ | additive | $18$ | \( 6358 = 2 \cdot 11 \cdot 17^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 43350 = 2 \cdot 3 \cdot 5^{2} \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 476850.gz
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 330.a4, its twist by $85$.
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 $0$.