Elliptic curve isogeny class with LMFDB label 85698.e (Cremona label 85698j)

• Label: 85698.e
• Number of curves: $4$
• Conductor: $85698$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 85698.e have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 + T\)
\(3\)	\(1\)
\(23\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 + 2 T + 7 T^{2}\)	1.7.c
\(11\)	\( 1 + 3 T + 11 T^{2}\)	1.11.d
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 3 T + 17 T^{2}\)	1.17.d
\(19\)	\( 1 - T + 19 T^{2}\)	1.19.ab
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 85698.e do not have complex multiplication.

Modular form 85698.2.a.e

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 85698.e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
85698.e1	85698j3	\([1, -1, 0, -569832, -165422656]\)	\(-189613868625/128\)	\(-13813524874368\)	\([]\)	\(498960\)	\(1.8371\)
85698.e2	85698j4	\([1, -1, 0, -450807, -236547235]\)	\(-1159088625/2097152\)	\(-18331984114873270272\)	\([]\)	\(1496880\)	\(2.3864\)
85698.e3	85698j2	\([1, -1, 0, -22317, 1350413]\)	\(-140625/8\)	\(-69930969676488\)	\([]\)	\(213840\)	\(1.4135\)
85698.e4	85698j1	\([1, -1, 0, 1488, 3050]\)	\(3375/2\)	\(-215836326162\)	\([]\)	\(71280\)	\(0.86416\)	\(\Gamma_0(N)\)-optimal