Elliptic curve isogeny class with Cremona label 49686ck (LMFDB label 49686.cn)

• Label: 49686ck
• Number of curves: $2$
• Conductor: $49686$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 49686ck have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1 + T\)
\(7\)	\(1\)
\(13\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - T + 5 T^{2}\)	1.5.ab
\(11\)	\( 1 + T + 11 T^{2}\)	1.11.b
\(17\)	\( 1 - 5 T + 17 T^{2}\)	1.17.af
\(19\)	\( 1 - T + 19 T^{2}\)	1.19.ab
\(23\)	\( 1 + 5 T + 23 T^{2}\)	1.23.f
\(29\)	\( 1 + 3 T + 29 T^{2}\)	1.29.d
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 49686ck do not have complex multiplication.

Modular form 49686.2.a.ck

Isogeny matrix

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 49686ck

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
49686.cn1	49686ck1	\([1, 1, 1, -3039, -114171]\)	\(-156116857/186624\)	\(-3710585458944\)	\([]\)	\(138240\)	\(1.1057\)	\(\Gamma_0(N)\)-optimal
49686.cn2	49686ck2	\([1, 1, 1, 25626, 2121699]\)	\(93603087383/150994944\)	\(-3002184304164864\)	\([]\)	\(414720\)	\(1.6550\)