Elliptic curve isogeny class with Cremona label 289296bf (LMFDB label 289296.bf)

• Label: 289296bf
• Number of curves: $4$
• Conductor: $289296$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 289296bf have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(7\)	\(1\)
\(41\)	\(1 - T\)

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 289296bf do not have complex multiplication.

Modular form 289296.2.a.bf

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 289296bf

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
289296.bf3	289296bf1	\([0, 0, 0, -63651, -5779550]\)	\(81182737/5904\)	\(2074063578660864\)	\([2]\)	\(1327104\)	\(1.6854\)	\(\Gamma_0(N)\)-optimal
289296.bf2	289296bf2	\([0, 0, 0, -204771, 28794850]\)	\(2703045457/544644\)	\(191332365131464704\)	\([2, 2]\)	\(2654208\)	\(2.0320\)
289296.bf1	289296bf3	\([0, 0, 0, -3097731, 2098418434]\)	\(9357915116017/538002\)	\(188999043605471232\)	\([2]\)	\(5308416\)	\(2.3786\)
289296.bf4	289296bf4	\([0, 0, 0, 430269, 171932866]\)	\(25076571983/50863698\)	\(-17868316988110675968\)	\([2]\)	\(5308416\)	\(2.3786\)