Elliptic curve isogeny class with Cremona label 49686cg (LMFDB label 49686.ca)

• Label: 49686cg
• Number of curves: $4$
• Conductor: $49686$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 49686cg 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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(11\)	\( 1 - 3 T + 11 T^{2}\)	1.11.ad
\(17\)	\( 1 - 3 T + 17 T^{2}\)	1.17.ad
\(19\)	\( 1 - 5 T + 19 T^{2}\)	1.19.af
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 9 T + 29 T^{2}\)	1.29.j
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 49686.2.a.cg

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 49686cg

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
49686.ca3	49686cg1	\([1, 1, 1, -468049, 119653295]\)	\(19968681097/628992\)	\(357185216579772672\)	\([4]\)	\(774144\)	\(2.1429\)	\(\Gamma_0(N)\)-optimal
49686.ca2	49686cg2	\([1, 1, 1, -1130529, -295589169]\)	\(281397674377/96589584\)	\(54850254821031340944\)	\([2, 2]\)	\(1548288\)	\(2.4895\)
49686.ca4	49686cg3	\([1, 1, 1, 3341211, -2050299945]\)	\(7264187703863/7406095788\)	\(-4205694075675560503308\)	\([2]\)	\(3096576\)	\(2.8361\)
49686.ca1	49686cg4	\([1, 1, 1, -16201949, -25103146489]\)	\(828279937799497/193444524\)	\(109851197155307273484\)	\([2]\)	\(3096576\)	\(2.8361\)