Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-28202180x+57653325447\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-28202180xz^2+57653325447z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-451234875x+3689361593750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(12251/4, -12255/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
|
| Discriminant: | $\Delta$ | = | $5805294315187500000$ | = | $2^{5} \cdot 3^{7} \cdot 5^{9} \cdot 7^{6} \cdot 19^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{14809006736693}{34656} \) | = | $2^{-5} \cdot 3^{-1} \cdot 13^{3} \cdot 19^{-2} \cdot 1889^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8429072983406395446882315939$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.11356764515335276548736310380$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.038451011044118$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.872604309363805$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.20721115907349946283395109002$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ 5\cdot2\cdot2\cdot2^{2}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $8.2884463629399785133580436009 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 8.288446363 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.207211 \cdot 1.000000 \cdot 160}{2^2} \\ & \approx 8.288446363\end{aligned}$$
Modular invariants
Modular form 418950.2.a.qc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 29491200 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2277 & 4 \\ 2276 & 5 \end{array}\right),\left(\begin{array}{rr} 1828 & 1 \\ 1823 & 0 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1139 & 0 \end{array}\right),\left(\begin{array}{rr} 1522 & 1 \\ 1519 & 0 \end{array}\right),\left(\begin{array}{rr} 1921 & 4 \\ 1562 & 9 \end{array}\right),\left(\begin{array}{rr} 1996 & 289 \\ 1425 & 856 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$363095654400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 | $4$ | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 46550 = 2 \cdot 5^{2} \cdot 7^{2} \cdot 19 \) |
| $5$ | additive | $14$ | \( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \) |
| $7$ | additive | $26$ | \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 22050 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 418950qc
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2850u2, its twist by $105$.
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 $0$.