Elliptic curve isogeny class with LMFDB label 21294.ck (Cremona label 21294cg)

• Label: 21294.ck
• Number of curves: $4$
• Conductor: $21294$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 21294.ck have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1\)
\(7\)	\(1 + T\)
\(13\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 21294.ck do not have complex multiplication.

Modular form 21294.2.a.ck

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 21294.ck

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
21294.ck1	21294cg4	\([1, -1, 1, -634289, -194274349]\)	\(8020417344913/187278\)	\(658983294072558\)	\([2]\)	\(258048\)	\(1.9558\)
21294.ck2	21294cg2	\([1, -1, 1, -41099, -2792617]\)	\(2181825073/298116\)	\(1048993815054276\)	\([2, 2]\)	\(129024\)	\(1.6092\)
21294.ck3	21294cg1	\([1, -1, 1, -10679, 383231]\)	\(38272753/4368\)	\(15369872748048\)	\([4]\)	\(64512\)	\(1.2626\)	\(\Gamma_0(N)\)-optimal
21294.ck4	21294cg3	\([1, -1, 1, 65371, -14930197]\)	\(8780064047/32388174\)	\(-113965685192682414\)	\([2]\)	\(258048\)	\(1.9558\)