Elliptic curve isogeny class with LMFDB label 113.a (Cremona label 113a)

• Label: 113.a
• Number of curves: $2$
• Conductor: $113$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 113.a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(113\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + T + 2 T^{2}\)	1.2.b
\(3\)	\( 1 - 2 T + 3 T^{2}\)	1.3.ac
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 - 6 T + 19 T^{2}\)	1.19.ag
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 113.a do not have complex multiplication.

Modular form 113.2.a.a

Isogeny matrix

The \((i,j)\)-th 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.

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

Isogeny graph

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 113.a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
113.a1	113a2	\([1, 1, 1, -2, -2]\)	\(912673/113\)	\(113\)	\([2]\)	\(12\)	\(-0.87617\)
113.a2	113a1	\([1, 1, 1, 3, -4]\)	\(2924207/12769\)	\(-12769\)	\([2]\)	\(6\)	\(-0.52960\)	\(\Gamma_0(N)\)-optimal