Elliptic curve with Cremona label 493680et2 (LMFDB label 493680.et2)

• Label: 493680et2
• Conductor: $493680$
• Discriminant: $1.493\times 10^{18}$
• j-invariant: $\frac{41948679809104}{3291890625}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{2}\Z \oplus \Z/{2}\Z$

Minimal Weierstrass equation

$y^2=x^3+x^2-556156x-148607956$

(, )

Mordell-Weil group structure

$\Z/{2}\Z \oplus \Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(-521, 0)$	$0$	$2$
$(854, 0)$	$0$	$2$

Integral points

$\left(-521, 0\right)$, $\left(-334, 0\right)$, $\left(854, 0\right)$

Invariants

Conductor:	$N$	=	$493680$	=	$2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17$
Discriminant:	$\Delta$	=	$1492936972164000000$	=	$2^{8} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6} \cdot 17^{2}$
j-invariant:	$j$	=	$\frac{41948679809104}{3291890625}$	=	$2^{4} \cdot 3^{-6} \cdot 5^{-6} \cdot 17^{-2} \cdot 13789^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$2.2312149013154122475012006736$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$0.57016914454293010252540747031$
$abc$ quality:	$Q$	≈	$0.9534609190750047$
Szpiro ratio:	$\sigma_{m}$	≈	$3.913151402430552$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$0.17569325736217492022099409171$
Tamagawa product:	$\prod_{p}c_p$	=	$192$  = $2\cdot( 2 \cdot 3 )\cdot2\cdot2^{2}\cdot2$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$4$
Special value:	$L(E,1)$	≈	$2.1083190883460990426519291005$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\displaystyle 2.108319088 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.175693 \cdot 1.000000 \cdot 192}{4^2} \approx 2.108319088$

Modular invariants

Modular form 493680.2.a.et

$q + q^{3} - q^{5} + q^{9} - 6 q^{13} - q^{15} + q^{17} + 4 q^{19} + O(q^{20})$

For more coefficients, see the Downloads section to the right.

Modular degree:	8847360
$\Gamma_0(N)$-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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$2$	$I_0^{*}$	additive	1	4	8	0
$3$	$6$	$I_{6}$	split multiplicative	-1	1	6	6
$5$	$2$	$I_{6}$	nonsplit multiplicative	1	1	6	6
$11$	$4$	$I_0^{*}$	additive	-1	2	6	0
$17$	$2$	$I_{2}$	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
$2$	2Cs	2.6.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $11220 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 17$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3301 & 5104 \\ 3542 & 10209 \end{array}\right),\left(\begin{array}{rr} 11217 & 4 \\ 11216 & 5 \end{array}\right),\left(\begin{array}{rr} 8977 & 8162 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8163 & 8162 \\ 10538 & 3059 \end{array}\right),\left(\begin{array}{rr} 8667 & 3058 \\ 10208 & 2045 \end{array}\right),\left(\begin{array}{rr} 4079 & 0 \\ 0 & 11219 \end{array}\right)$.

The torsion field $K:=\Q(E[11220])$ is a degree-$47648342016000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11220\Z)$.

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	$2$	$121 = 11^{2}$
$3$	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.
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 2040j2, 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 $0$.