Elliptic curve isogeny class with Cremona label 432a (LMFDB label 432.e)

• Label: 432a
• Number of curves: $4$
• Conductor: $432$
• CM: \(\Q(\sqrt{-3}) \)
• Rank: $0$

Rank

The elliptic curves in class 432a 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.e
\(7\)	\( 1 - 3 T + 7 T^{2}\)	1.7.ad
\(11\)	\( 1 - 4 T + 11 T^{2}\)	1.11.ae
\(13\)	\( 1 - T + 13 T^{2}\)	1.13.ab
\(17\)	\( 1 - 4 T + 17 T^{2}\)	1.17.ae
\(19\)	\( 1 - T + 19 T^{2}\)	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 432a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 432.2.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.

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

Isogeny graph

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

Elliptic curves in class 432a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
432.e3	432a1	\([0, 0, 0, 0, -16]\)	\(0\)	\(-110592\)	\([]\)	\(24\)	\(-0.35332\)	\(\Gamma_0(N)\)-optimal	\(-3\)
432.e2	432a2	\([0, 0, 0, -480, -4048]\)	\(-12288000\)	\(-995328\)	\([]\)	\(72\)	\(0.19599\)		\(-27\)
432.e4	432a3	\([0, 0, 0, 0, 432]\)	\(0\)	\(-80621568\)	\([]\)	\(72\)	\(0.19599\)		\(-3\)
432.e1	432a4	\([0, 0, 0, -4320, 109296]\)	\(-12288000\)	\(-725594112\)	\([]\)	\(216\)	\(0.74530\)		\(-27\)