Elliptic curve isogeny class with Cremona label 85176f (LMFDB label 85176.c)

• Label: 85176f
• Number of curves: $2$
• Conductor: $85176$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 85176f have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(7\)	\(1 + T\)
\(13\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 3 T + 5 T^{2}\)	1.5.d
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(19\)	\( 1 - 7 T + 19 T^{2}\)	1.19.ah
\(23\)	\( 1 - T + 23 T^{2}\)	1.23.ab
\(29\)	\( 1 + T + 29 T^{2}\)	1.29.b
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 85176f do not have complex multiplication.

Modular form 85176.2.a.f

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.

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

Isogeny graph

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

Elliptic curves in class 85176f

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
85176.c1	85176f1	\([0, 0, 0, -132327, 18388890]\)	\(10536048/91\)	\(2213261675718912\)	\([2]\)	\(903168\)	\(1.7688\)	\(\Gamma_0(N)\)-optimal
85176.c2	85176f2	\([0, 0, 0, -41067, 43302870]\)	\(-78732/8281\)	\(-805627249961683968\)	\([2]\)	\(1806336\)	\(2.1154\)