Elliptic curve isogeny class with LMFDB label 397488.fk (Cremona label 397488fk)

• Label: 397488.fk
• Number of curves: $2$
• Conductor: $397488$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 397488.fk have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 397488.fk do not have complex multiplication.

Modular form 397488.2.a.fk

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 397488.fk

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
397488.fk1	397488fk1	\([0, 1, 0, -24963605, -48015742566]\)	\(416013434950254592/771895089\)	\(3192247514571150672\)	\([2]\)	\(26542080\)	\(2.8066\)	\(\Gamma_0(N)\)-optimal
397488.fk2	397488fk2	\([0, 1, 0, -24705620, -49056454056]\)	\(-25203028990703632/1121144263281\)	\(-74185625252697541809408\)	\([2]\)	\(53084160\)	\(3.1532\)