Elliptic curve isogeny class with LMFDB label 1728.o (Cremona label 1728v)

• Label: 1728.o
• Number of curves: $4$
• Conductor: $1728$
• CM: \(\Q(\sqrt{-3}) \)
• Rank: $1$

Elliptic curves in class 1728.o

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
1728.o1	1728v4	\([0, 0, 0, -1080, 13662]\)	\(-12288000\)	\(-11337408\)	\([]\)	\(432\)	\(0.39872\)		\(-27\)
1728.o2	1728v2	\([0, 0, 0, -120, -506]\)	\(-12288000\)	\(-15552\)	\([]\)	\(144\)	\(-0.15058\)		\(-27\)
1728.o3	1728v1	\([0, 0, 0, 0, -2]\)	\(0\)	\(-1728\)	\([]\)	\(48\)	\(-0.69989\)	\(\Gamma_0(N)\)-optimal	\(-3\)
1728.o4	1728v3	\([0, 0, 0, 0, 54]\)	\(0\)	\(-1259712\)	\([]\)	\(144\)	\(-0.15058\)		\(-3\)

Rank

The elliptic curves in class 1728.o have rank \(1\).

Complex multiplication

Each elliptic curve in class 1728.o has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 1728.2.a.o

Isogeny matrix

The \(i,j\) 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.

\(\left(\begin{array}{rrrr} 1 & 27 & 9 & 3 \\ 27 & 1 & 3 & 9 \\ 9 & 3 & 1 & 3 \\ 3 & 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.