Elliptic curve isogeny class with LMFDB label 171.b (Cremona label 171b)

• Label: 171.b
• Number of curves: $3$
• Conductor: $171$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 171.b have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1\)
\(19\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T^{2}\)	1.2.a
\(5\)	\( 1 + 3 T + 5 T^{2}\)	1.5.d
\(7\)	\( 1 + T + 7 T^{2}\)	1.7.b
\(11\)	\( 1 + 3 T + 11 T^{2}\)	1.11.d
\(13\)	\( 1 + 4 T + 13 T^{2}\)	1.13.e
\(17\)	\( 1 - 3 T + 17 T^{2}\)	1.17.ad
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 171.b do not have complex multiplication.

Modular form 171.2.a.b

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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 171.b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
171.b1	171b3	\([0, 0, 1, -6924, 221760]\)	\(-50357871050752/19\)	\(-13851\)	\([3]\)	\(72\)	\(0.58275\)
171.b2	171b2	\([0, 0, 1, -84, 315]\)	\(-89915392/6859\)	\(-5000211\)	\([3]\)	\(24\)	\(0.033439\)
171.b3	171b1	\([0, 0, 1, 6, 0]\)	\(32768/19\)	\(-13851\)	\([]\)	\(8\)	\(-0.51587\)	\(\Gamma_0(N)\)-optimal