Elliptic curve isogeny class with LMFDB label 21168.ce (Cremona label 21168cz)

• Label: 21168.ce
• Number of curves: $2$
• Conductor: $21168$
• CM: \(\Q(\sqrt{-3}) \)
• Rank: $0$

Rank

The elliptic curves in class 21168.ce have rank \(0\).

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
\(13\)	\( 1 - 7 T + 13 T^{2}\)	1.13.ah
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 - 8 T + 19 T^{2}\)	1.19.ai
\(23\)	\( 1 + 23 T^{2}\)	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 21168.ce has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 21168.2.a.ce

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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 21168.ce

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
21168.ce1	21168cz2	\([0, 0, 0, 0, -189]\)	\(0\)	\(-15431472\)	\([]\)	\(3888\)	\(0.058209\)		\(-3\)
21168.ce2	21168cz1	\([0, 0, 0, 0, 7]\)	\(0\)	\(-21168\)	\([]\)	\(1296\)	\(-0.49110\)	\(\Gamma_0(N)\)-optimal	\(-3\)