Elliptic curve isogeny class with LMFDB label 48.a (Cremona label 48a)

• Label: 48.a
• Number of curves: $6$
• Conductor: $48$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 48.a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 4 T + 11 T^{2}\)	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.ae
\(23\)	\( 1 - 8 T + 23 T^{2}\)	1.23.ai
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 48.a do not have complex multiplication.

Modular form 48.2.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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 48.a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
48.a1	48a5	\([0, 1, 0, -384, 2772]\)	\(3065617154/9\)	\(18432\)	\([4]\)	\(8\)	\(0.047795\)
48.a2	48a2	\([0, 1, 0, -64, -220]\)	\(28756228/3\)	\(3072\)	\([2]\)	\(4\)	\(-0.29878\)
48.a3	48a3	\([0, 1, 0, -24, 36]\)	\(1556068/81\)	\(82944\)	\([2, 4]\)	\(4\)	\(-0.29878\)
48.a4	48a1	\([0, 1, 0, -4, -4]\)	\(35152/9\)	\(2304\)	\([2, 2]\)	\(2\)	\(-0.64535\)	\(\Gamma_0(N)\)-optimal
48.a5	48a4	\([0, 1, 0, 1, 0]\)	\(2048/3\)	\(-48\)	\([2]\)	\(4\)	\(-0.99193\)
48.a6	48a6	\([0, 1, 0, 16, 180]\)	\(207646/6561\)	\(-13436928\)	\([8]\)	\(8\)	\(0.047795\)