Elliptic curve isogeny class with Cremona label 69696eu (LMFDB label 69696.gu)

• Label: 69696eu
• Number of curves: $2$
• Conductor: $69696$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 69696eu have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(11\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - T + 5 T^{2}\)	1.5.ab
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(17\)	\( 1 + 4 T + 17 T^{2}\)	1.17.e
\(19\)	\( 1 + 6 T + 19 T^{2}\)	1.19.g
\(23\)	\( 1 - 3 T + 23 T^{2}\)	1.23.ad
\(29\)	\( 1 - 4 T + 29 T^{2}\)	1.29.ae
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 69696eu do not have complex multiplication.

Modular form 69696.2.a.eu

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.

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

Isogeny graph

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

Elliptic curves in class 69696eu

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
69696.gu2	69696eu1	\([0, 0, 0, -13068, -106480]\)	\(19683/11\)	\(137928013774848\)	\([2]\)	\(245760\)	\(1.4034\)	\(\Gamma_0(N)\)-optimal
69696.gu1	69696eu2	\([0, 0, 0, -129228, 17782160]\)	\(19034163/121\)	\(1517208151523328\)	\([2]\)	\(491520\)	\(1.7500\)