Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2+554624x-667120060\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z+554624xz^2-667120060z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+44924517x-486465297318\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(667, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([667:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6006, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(667, 0\right) \)
\([667:0:1]\)
\( \left(667, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 493680 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-203286624750000000000$ | = | $-1 \cdot 2^{10} \cdot 3^{3} \cdot 5^{12} \cdot 11^{6} \cdot 17 $ |
|
| j-invariant: | $j$ | = | \( \frac{10400706415004}{112060546875} \) | = | $2^{2} \cdot 3^{-3} \cdot 5^{-12} \cdot 17^{-1} \cdot 13751^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5777884915953849022098167343$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.80121820472957853899781817744$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.000625309766843$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.131360530230603$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.087846628681087460110497045855$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2^{2}\cdot3\cdot2\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.1083190883460990426519291005 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.108319088 \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 0.087847 \cdot 1.000000 \cdot 96}{2^2} \\ & \approx 2.108319088\end{aligned}$$
Modular invariants
Modular form 493680.2.a.et
For more coefficients, see the Downloads section to the right.
| Modular degree: | 17694720 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
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
$\left(\begin{array}{rr} 7 & 6 \\ 22434 & 22435 \end{array}\right),\left(\begin{array}{rr} 20659 & 20658 \\ 10978 & 5875 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 19724 & 4081 \\ 19063 & 20406 \end{array}\right),\left(\begin{array}{rr} 8977 & 10208 \\ 7348 & 18393 \end{array}\right),\left(\begin{array}{rr} 22433 & 8 \\ 22432 & 9 \end{array}\right),\left(\begin{array}{rr} 4079 & 0 \\ 0 & 22439 \end{array}\right),\left(\begin{array}{rr} 19117 & 21164 \\ 12958 & 16059 \end{array}\right),\left(\begin{array}{rr} 10088 & 12243 \\ 3245 & 4082 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 6171 = 3 \cdot 11^{2} \cdot 17 \) |
| $3$ | split multiplicative | $4$ | \( 32912 = 2^{4} \cdot 11^{2} \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 98736 = 2^{4} \cdot 3 \cdot 11^{2} \cdot 17 \) |
| $11$ | additive | $62$ | \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 29040 = 2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 493680et
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2040j4, its twist by $44$.
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 $0$.