Elliptic curve isogeny class with LMFDB label 576.f (Cremona label 576e)

• Label: 576.f
• Number of curves: $4$
• Conductor: $576$
• CM: \(\Q(\sqrt{-3}) \)
• Rank: $0$

Rank

The elliptic curves in class 576.f 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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 - 4 T + 7 T^{2}\)	1.7.ae
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 - 8 T + 19 T^{2}\)	1.19.ai
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 576.f has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 576.2.a.f

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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 576.f

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
576.f1	576e4	\([0, 0, 0, -540, 4752]\)	\(54000\)	\(322486272\)	\([2]\)	\(192\)	\(0.42704\)		\(-12\)
576.f2	576e2	\([0, 0, 0, -60, -176]\)	\(54000\)	\(442368\)	\([2]\)	\(64\)	\(-0.12227\)		\(-12\)
576.f3	576e1	\([0, 0, 0, 0, -8]\)	\(0\)	\(-27648\)	\([2]\)	\(32\)	\(-0.46884\)	\(\Gamma_0(N)\)-optimal	\(-3\)
576.f4	576e3	\([0, 0, 0, 0, 216]\)	\(0\)	\(-20155392\)	\([2]\)	\(96\)	\(0.080464\)		\(-3\)