Elliptic curve isogeny class with LMFDB label 27.a (Cremona label 27a)

• Label: 27.a
• Number of curves: $4$
• Conductor: $27$
• CM: \(\Q(\sqrt{-3}) \)
• Rank: $0$

Rank

The elliptic curves in class 27.a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T^{2}\)	1.2.a
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 + T + 7 T^{2}\)	1.7.b
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 5 T + 13 T^{2}\)	1.13.af
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 7 T + 19 T^{2}\)	1.19.h
\(23\)	\( 1 + 23 T^{2}\)	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 27.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 27.2.a.a

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 27.a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
27.a1	27a2	\([0, 0, 1, -270, -1708]\)	\(-12288000\)	\(-177147\)	\([]\)	\(3\)	\(0.052148\)		\(-27\)
27.a2	27a4	\([0, 0, 1, -30, 63]\)	\(-12288000\)	\(-243\)	\([3]\)	\(9\)	\(-0.49716\)		\(-27\)
27.a3	27a1	\([0, 0, 1, 0, -7]\)	\(0\)	\(-19683\)	\([3]\)	\(1\)	\(-0.49716\)	\(\Gamma_0(N)\)-optimal	\(-3\)
27.a4	27a3	\([0, 0, 1, 0, 0]\)	\(0\)	\(-27\)	\([3]\)	\(3\)	\(-1.0465\)		\(-3\)