Elliptic curve isogeny class with LMFDB label 49686.by (Cremona label 49686cq)

• Label: 49686.by
• Number of curves: $4$
• Conductor: $49686$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 49686.by have rank \(0\).

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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 - 6 T + 19 T^{2}\)	1.19.ag
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 10 T + 29 T^{2}\)	1.29.k
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 49686.2.a.by

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 49686.by

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
49686.by1	49686cq4	\([1, 1, 1, -587922189, -5487155647485]\)	\(18013780041269221/9216\)	\(11497962209901253632\)	\([2]\)	\(11232000\)	\(3.4254\)
49686.by2	49686cq3	\([1, 1, 1, -36738829, -85779192829]\)	\(-4395631034341/3145728\)	\(-3924637767646294573056\)	\([2]\)	\(5616000\)	\(3.0789\)
49686.by3	49686cq2	\([1, 1, 1, -1751604, 336963297]\)	\(476379541/236196\)	\(294680195543602051092\)	\([2]\)	\(2246400\)	\(2.6207\)
49686.by4	49686cq1	\([1, 1, 1, 401456, 40702241]\)	\(5735339/3888\)	\(-4850702807302091376\)	\([2]\)	\(1123200\)	\(2.2741\)	\(\Gamma_0(N)\)-optimal