Elliptic curve isogeny class with LMFDB label 31734.f (Cremona label 31734f)

• Label: 31734.f
• Number of curves: $2$
• Conductor: $31734$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 31734.f have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 + T\)
\(3\)	\(1\)
\(41\)	\(1 - T\)
\(43\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 + T + 7 T^{2}\)	1.7.b
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 4 T + 13 T^{2}\)	1.13.e
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 - 3 T + 23 T^{2}\)	1.23.ad
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 31734.f do not have complex multiplication.

Modular form 31734.2.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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 31734.f

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
31734.f1	31734f1	\([1, -1, 0, -3537, -80163]\)	\(-6713831364625/7221248\)	\(-5264289792\)	\([]\)	\(20160\)	\(0.78142\)	\(\Gamma_0(N)\)-optimal
31734.f2	31734f2	\([1, -1, 0, 4383, -369171]\)	\(12771880859375/87675231152\)	\(-63915243509808\)	\([3]\)	\(60480\)	\(1.3307\)