Elliptic curve with Cremona label 406272l1 (LMFDB label 406272.l1)

• Label: 406272l1
• Conductor: $406272$
• Discriminant: $682149376512$
• j-invariant: $\frac{2744000}{9}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{2}\Z$

Minimal Weierstrass equation

$y^2=x^3-x^2-12343x-522221$

(, )

Mordell-Weil group structure

$\Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(-61, 0)$	$0$	$2$

Integral points

$\left(-61, 0\right)$

Invariants

Conductor:	$N$	=	$406272$	=	$2^{8} \cdot 3 \cdot 23^{2}$
Discriminant:	$\Delta$	=	$682149376512$	=	$2^{9} \cdot 3^{2} \cdot 23^{6}$
j-invariant:	$j$	=	$\frac{2744000}{9}$	=	$2^{6} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$1.1360326835814698127834608197$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.95157480980306401468283968730$
$abc$ quality:	$Q$	≈	$1.106715441314673$
Szpiro ratio:	$\sigma_{m}$	≈	$3.08764244177971$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$0.45303012744484150253174394407$
Tamagawa product:	$\prod_{p}c_p$	=	$16$  = $2\cdot2\cdot2^{2}$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$2$
Special value:	$L(E,1)$	≈	$1.8121205097793660101269757763$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 1.812120510 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.453030 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 1.812120510\end{aligned}$

Modular invariants

Modular form 406272.2.a.l

$q - q^{3} + 4 q^{7} + q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})$

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

Modular degree:	720896
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$2$	$III$	additive	1	8	9	0
$3$	$2$	$I_{2}$	nonsplit multiplicative	1	1	2	2
$23$	$4$	$I_0^{*}$	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	8.24.0.130

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $1104 = 2^{4} \cdot 3 \cdot 23$, index $96$, genus $1$, and generators

$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 8 \\ 976 & 1011 \end{array}\right),\left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 898 & 253 \\ 1035 & 346 \end{array}\right),\left(\begin{array}{rr} 1007 & 0 \\ 0 & 1103 \end{array}\right),\left(\begin{array}{rr} 1089 & 16 \\ 1088 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 12 \\ 1004 & 995 \end{array}\right),\left(\begin{array}{rr} 137 & 230 \\ 874 & 1011 \end{array}\right),\left(\begin{array}{rr} 737 & 874 \\ 46 & 93 \end{array}\right)$.

The torsion field $K:=\Q(E[1104])$ is a degree-$3282960384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1104\Z)$.

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	$4$	$529 = 23^{2}$
$3$	nonsplit multiplicative	$4$	$135424 = 2^{8} \cdot 23^{2}$
$23$	additive	$266$	$768 = 2^{8} \cdot 3$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 406272l consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 768a1, its twist by $-23$.

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 $0$.