Elliptic curve isogeny class with Cremona label 129600ef (LMFDB label 129600.cd)

• Label: 129600ef
• Number of curves: $4$
• Conductor: $129600$
• CM: no
• Rank: $2$

Rank

The elliptic curves in class 129600ef have rank \(2\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(5\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 + T + 7 T^{2}\)	1.7.b
\(11\)	\( 1 - 6 T + 11 T^{2}\)	1.11.ag
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 - 9 T + 23 T^{2}\)	1.23.aj
\(29\)	\( 1 + 3 T + 29 T^{2}\)	1.29.d
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 129600ef do not have complex multiplication.

Modular form 129600.2.a.ef

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.

\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 129600ef

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
129600.cd4	129600ef1	\([0, 0, 0, 4500, -18000]\)	\(3375/2\)	\(-5971968000000\)	\([]\)	\(165888\)	\(1.1409\)	\(\Gamma_0(N)\)-optimal
129600.cd3	129600ef2	\([0, 0, 0, -67500, -7074000]\)	\(-140625/8\)	\(-1934917632000000\)	\([]\)	\(497664\)	\(1.6902\)
129600.cd1	129600ef3	\([0, 0, 0, -1723500, 870894000]\)	\(-189613868625/128\)	\(-382205952000000\)	\([]\)	\(1161216\)	\(2.1138\)
129600.cd2	129600ef4	\([0, 0, 0, -1363500, 1244862000]\)	\(-1159088625/2097152\)	\(-507227047723008000000\)	\([]\)	\(3483648\)	\(2.6631\)