Elliptic curve isogeny class with LMFDB label 3420.f (Cremona label 3420a)

• Label: 3420.f
• Number of curves: $2$
• Conductor: $3420$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 3420.f have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(5\)	\(1 - T\)
\(19\)	\(1 + T\)

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 3420.f do not have complex multiplication.

Modular form 3420.2.a.f

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 LMFDB numbering.

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

Isogeny graph

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

Elliptic curves in class 3420.f

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
3420.f1	3420a1	\([0, 0, 0, -8292, -271051]\)	\(5405726654464/407253125\)	\(4750200450000\)	\([2]\)	\(5760\)	\(1.1777\)	\(\Gamma_0(N)\)-optimal
3420.f2	3420a2	\([0, 0, 0, 7953, -1203514]\)	\(298091207216/3525390625\)	\(-657922500000000\)	\([2]\)	\(11520\)	\(1.5243\)