Elliptic curve isogeny class with Cremona label 34496cf (LMFDB label 34496.bz)

• Label: 34496cf
• Number of curves: $4$
• Conductor: $34496$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 34496cf have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(7\)	\(1\)
\(11\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 - T + 3 T^{2}\)	1.3.ab
\(5\)	\( 1 + 3 T + 5 T^{2}\)	1.5.d
\(13\)	\( 1 - 6 T + 13 T^{2}\)	1.13.ag
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 6 T + 19 T^{2}\)	1.19.g
\(23\)	\( 1 + 3 T + 23 T^{2}\)	1.23.d
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 34496cf do not have complex multiplication.

Modular form 34496.2.a.cf

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 34496cf

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
34496.bz4	34496cf1	\([0, 0, 0, -11564, -15745072]\)	\(-5545233/3469312\)	\(-106996980118454272\)	\([2]\)	\(221184\)	\(1.9467\)	\(\Gamma_0(N)\)-optimal
34496.bz3	34496cf2	\([0, 0, 0, -1015084, -389455920]\)	\(3750606459153/45914176\)	\(1416038158755168256\)	\([2, 2]\)	\(442368\)	\(2.2933\)
34496.bz2	34496cf3	\([0, 0, 0, -1893164, 385010640]\)	\(24331017010833/12004097336\)	\(370218119327400329216\)	\([2]\)	\(884736\)	\(2.6399\)
34496.bz1	34496cf4	\([0, 0, 0, -16193324, -25081416752]\)	\(15226621995131793/2324168\)	\(71679617540292608\)	\([2]\)	\(884736\)	\(2.6399\)