Elliptic curve isogeny class with Cremona label 103968ca (LMFDB label 103968.bz)

• Label: 103968ca
• Number of curves: $4$
• Conductor: $103968$
• CM: no
• Rank: $2$

Rank

The elliptic curves in class 103968ca have rank \(2\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(19\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 - 6 T + 11 T^{2}\)	1.11.ag
\(13\)	\( 1 - 4 T + 13 T^{2}\)	1.13.ae
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 103968ca do not have complex multiplication.

Modular form 103968.2.a.ca

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 Cremona numbering.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 103968ca

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
103968.bz3	103968ca1	\([0, 0, 0, -371469, -86286220]\)	\(2582630848/29241\)	\(64183194496512576\)	\([2, 2]\)	\(1105920\)	\(2.0385\)	\(\Gamma_0(N)\)-optimal
103968.bz4	103968ca2	\([0, 0, 0, -79059, -218514022]\)	\(-3112136/1172889\)	\(-20595673967325368832\)	\([2]\)	\(2211840\)	\(2.3851\)
103968.bz2	103968ca3	\([0, 0, 0, -680124, 77918240]\)	\(247673152/124659\)	\(17511877908943220736\)	\([2]\)	\(2211840\)	\(2.3851\)
103968.bz1	103968ca4	\([0, 0, 0, -5927259, -5554294738]\)	\(1311494070536/171\)	\(3002722549544448\)	\([2]\)	\(2211840\)	\(2.3851\)