Elliptic curve isogeny class with Cremona label 314330bf (LMFDB label 314330.bf)

• Label: 314330bf
• Number of curves: $4$
• Conductor: $314330$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 314330bf have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(5\)	\(1 - T\)
\(17\)	\(1 - T\)
\(43\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 - 2 T + 3 T^{2}\)	1.3.ac
\(7\)	\( 1 + 2 T + 7 T^{2}\)	1.7.c
\(11\)	\( 1 - 6 T + 11 T^{2}\)	1.11.ag
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(19\)	\( 1 + 8 T + 19 T^{2}\)	1.19.i
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 314330bf do not have complex multiplication.

Modular form 314330.2.a.bf

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 314330bf

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
314330.bf2	314330bf1	\([1, 1, 1, -4721460, -3950685835]\)	\(1841373668746009/31443200\)	\(198763882622316800\)	\([2]\)	\(12902400\)	\(2.4488\)	\(\Gamma_0(N)\)-optimal
314330.bf3	314330bf2	\([1, 1, 1, -4573540, -4209605003]\)	\(-1673672305534489/241375690000\)	\(-1525823367692878810000\)	\([2]\)	\(25804800\)	\(2.7953\)
314330.bf1	314330bf3	\([1, 1, 1, -7707595, 1616852857]\)	\(8010684753304969/4456448000000\)	\(28170825716989952000000\)	\([2]\)	\(38707200\)	\(2.9981\)
314330.bf4	314330bf4	\([1, 1, 1, 30159925, 12840785785]\)	\(479958568556831351/289000000000000\)	\(-1826873921161000000000000\)	\([2]\)	\(77414400\)	\(3.3447\)