Elliptic curve isogeny class with Cremona label 296769f (LMFDB label 296769.f)

• Label: 296769f
• Number of curves: $4$
• Conductor: $296769$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 296769f have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1 - T\)
\(11\)	\(1 + T\)
\(17\)	\(1 + T\)
\(23\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + T + 2 T^{2}\)	1.2.b
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(19\)	\( 1 + 8 T + 19 T^{2}\)	1.19.i
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 296769f do not have complex multiplication.

Modular form 296769.2.a.f

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 296769f

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
296769.f3	296769f1	\([1, 0, 0, -6359, -195216]\)	\(192100033/561\)	\(83048133729\)	\([2]\)	\(360448\)	\(0.96566\)	\(\Gamma_0(N)\)-optimal
296769.f2	296769f2	\([1, 0, 0, -9004, -18001]\)	\(545338513/314721\)	\(46590003021969\)	\([2, 2]\)	\(720896\)	\(1.3122\)
296769.f1	296769f3	\([1, 0, 0, -96289, 11451248]\)	\(666940371553/2756193\)	\(408015481010577\)	\([2]\)	\(1441792\)	\(1.6588\)
296769.f4	296769f4	\([1, 0, 0, 35961, -134910]\)	\(34741712447/20160657\)	\(-2984500781819073\)	\([2]\)	\(1441792\)	\(1.6588\)