Elliptic curve isogeny class with LMFDB label 288.b (Cremona label 288b)

• Label: 288.b
• Number of curves: $4$
• Conductor: $288$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 288.b have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 288.b do not have complex multiplication.

Modular form 288.2.a.b

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.

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

Isogeny graph

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

Elliptic curves in class 288.b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
288.b1	288b2	\([0, 0, 0, -291, -1910]\)	\(7301384/3\)	\(1119744\)	\([2]\)	\(64\)	\(0.12309\)
288.b2	288b3	\([0, 0, 0, -156, 736]\)	\(140608/3\)	\(8957952\)	\([4]\)	\(64\)	\(0.12309\)
288.b3	288b1	\([0, 0, 0, -21, -20]\)	\(21952/9\)	\(419904\)	\([2, 2]\)	\(32\)	\(-0.22349\)	\(\Gamma_0(N)\)-optimal
288.b4	288b4	\([0, 0, 0, 69, -146]\)	\(97336/81\)	\(-30233088\)	\([2]\)	\(64\)	\(0.12309\)