Elliptic curve isogeny class with LMFDB label 78650.bf (Cremona label 78650u)

• Label: 78650.bf
• Number of curves: $4$
• Conductor: $78650$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 78650.bf have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 + T\)
\(5\)	\(1\)
\(11\)	\(1\)
\(13\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 - 2 T + 3 T^{2}\)	1.3.ac
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 + 2 T + 19 T^{2}\)	1.19.c
\(23\)	\( 1 + 6 T + 23 T^{2}\)	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 78650.bf do not have complex multiplication.

Modular form 78650.2.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 LMFDB numbering.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 78650.bf

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
78650.bf1	78650u3	\([1, 1, 0, -627750, 183492500]\)	\(988345570681/44994560\)	\(1245478245440000000\)	\([2]\)	\(1866240\)	\(2.2341\)
78650.bf2	78650u1	\([1, 1, 0, -98375, -11846875]\)	\(3803721481/26000\)	\(719696656250000\)	\([2]\)	\(622080\)	\(1.6848\)	\(\Gamma_0(N)\)-optimal
78650.bf3	78650u2	\([1, 1, 0, -37875, -26185375]\)	\(-217081801/10562500\)	\(-292376766601562500\)	\([2]\)	\(1244160\)	\(2.0314\)
78650.bf4	78650u4	\([1, 1, 0, 340250, 699436500]\)	\(157376536199/7722894400\)	\(-213774664471225000000\)	\([2]\)	\(3732480\)	\(2.5807\)