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

• Label: 288.e
• Number of curves: $2$
• Conductor: $288$
• CM: \(\Q(\sqrt{-1}) \)
• Rank: $0$

Rank

The elliptic curves in class 288.e have rank \(0\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 4 T + 5 T^{2}\)	1.5.ae
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(17\)	\( 1 - 8 T + 17 T^{2}\)	1.17.ai
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 4 T + 29 T^{2}\)	1.29.e
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 288.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 288.2.a.e

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 288.e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
288.e1	288e2	\([0, 0, 0, -108, 0]\)	\(1728\)	\(80621568\)	\([2]\)	\(96\)	\(0.20657\)		\(-4\)
288.e2	288e1	\([0, 0, 0, 27, 0]\)	\(1728\)	\(-1259712\)	\([2]\)	\(48\)	\(-0.14000\)	\(\Gamma_0(N)\)-optimal	\(-4\)