Elliptic curve isogeny class with Cremona label 1728a (LMFDB label 1728.n)

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

Rank

The elliptic curves in class 1728a 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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 4 T + 5 T^{2}\)	1.5.e
\(7\)	\( 1 + 3 T + 7 T^{2}\)	1.7.d
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 + T + 13 T^{2}\)	1.13.b
\(17\)	\( 1 + 4 T + 17 T^{2}\)	1.17.e
\(19\)	\( 1 - T + 19 T^{2}\)	1.19.ab
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 1728.2.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.

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

Isogeny graph

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

Elliptic curves in class 1728a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
1728.n4	1728a1	\([0, 0, 0, 0, 2]\)	\(0\)	\(-1728\)	\([]\)	\(48\)	\(-0.69989\)	\(\Gamma_0(N)\)-optimal	\(-3\)
1728.n3	1728a2	\([0, 0, 0, 0, -54]\)	\(0\)	\(-1259712\)	\([]\)	\(144\)	\(-0.15058\)		\(-3\)
1728.n2	1728a3	\([0, 0, 0, -120, 506]\)	\(-12288000\)	\(-15552\)	\([]\)	\(144\)	\(-0.15058\)		\(-27\)
1728.n1	1728a4	\([0, 0, 0, -1080, -13662]\)	\(-12288000\)	\(-11337408\)	\([]\)	\(432\)	\(0.39872\)		\(-27\)