Elliptic curve isogeny class with LMFDB label 80.a (Cremona label 80a)

• Label: 80.a
• Number of curves: $4$
• Conductor: $80$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 80.a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(5\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(7\)	\( 1 - 4 T + 7 T^{2}\)	1.7.ae
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 80.a do not have complex multiplication.

Modular form 80.2.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.

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

Isogeny graph

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

Elliptic curves in class 80.a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
80.a1	80a3	\([0, 0, 0, -107, 426]\)	\(132304644/5\)	\(5120\)	\([4]\)	\(8\)	\(-0.20221\)
80.a2	80a1	\([0, 0, 0, -7, 6]\)	\(148176/25\)	\(6400\)	\([2, 2]\)	\(4\)	\(-0.54879\)	\(\Gamma_0(N)\)-optimal
80.a3	80a2	\([0, 0, 0, -2, -1]\)	\(55296/5\)	\(80\)	\([2]\)	\(8\)	\(-0.89536\)
80.a4	80a4	\([0, 0, 0, 13, 34]\)	\(237276/625\)	\(-640000\)	\([4]\)	\(8\)	\(-0.20221\)