Elliptic curve isogeny class with Cremona label 32a (LMFDB label 32.a)

• Label: 32a
• Number of curves: $4$
• Conductor: $32$
• CM: \(\Q(\sqrt{-1}) \)
• Rank: $0$

Rank

The elliptic curves in class 32a have rank \(0\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 6 T + 13 T^{2}\)	1.13.ag
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 10 T + 29 T^{2}\)	1.29.k
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 32a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 32.2.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 Cremona 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 Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 32a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
32.a4	32a1	\([0, 0, 0, 4, 0]\)	\(1728\)	\(-4096\)	\([4]\)	\(1\)	\(-0.61739\)	\(\Gamma_0(N)\)-optimal	\(-4\)
32.a3	32a2	\([0, 0, 0, -1, 0]\)	\(1728\)	\(64\)	\([2, 2]\)	\(2\)	\(-0.96396\)		\(-4\)
32.a1	32a3	\([0, 0, 0, -11, -14]\)	\(287496\)	\(512\)	\([2]\)	\(4\)	\(-0.61739\)		\(-16\)
32.a2	32a4	\([0, 0, 0, -11, 14]\)	\(287496\)	\(512\)	\([4]\)	\(4\)	\(-0.61739\)		\(-16\)