Elliptic curve isogeny class with LMFDB label 1764.e (Cremona label 1764b)

• Label: 1764.e
• Number of curves: $4$
• Conductor: $1764$
• CM: \(\Q(\sqrt{-3}) \)
• Rank: $1$

Rank

The elliptic curves in class 1764.e have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(7\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 8 T + 19 T^{2}\)	1.19.i
\(23\)	\( 1 + 23 T^{2}\)	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 1764.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 1764.2.a.e

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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 1764.e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
1764.e1	1764b4	\([0, 0, 0, -6615, 203742]\)	\(54000\)	\(592815428352\)	\([2]\)	\(1728\)	\(1.0534\)		\(-12\)
1764.e2	1764b2	\([0, 0, 0, -735, -7546]\)	\(54000\)	\(813189888\)	\([2]\)	\(576\)	\(0.50411\)		\(-12\)
1764.e3	1764b1	\([0, 0, 0, 0, -343]\)	\(0\)	\(-50824368\)	\([2]\)	\(288\)	\(0.15754\)	\(\Gamma_0(N)\)-optimal	\(-3\)
1764.e4	1764b3	\([0, 0, 0, 0, 9261]\)	\(0\)	\(-37050964272\)	\([2]\)	\(864\)	\(0.70685\)		\(-3\)