Elliptic curve isogeny class with Cremona label 17a (LMFDB label 17.a)

• Label: 17a
• Number of curves: $4$
• Conductor: $17$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 17a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(17\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + T + 2 T^{2}\)	1.2.b
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(7\)	\( 1 - 4 T + 7 T^{2}\)	1.7.ae
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 17a do not have complex multiplication.

Modular form 17.2.a.a

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 17a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
17.a3	17a1	\([1, -1, 1, -1, -14]\)	\(-35937/83521\)	\(-83521\)	\([4]\)	\(1\)	\(-0.37664\)	\(\Gamma_0(N)\)-optimal
17.a2	17a2	\([1, -1, 1, -6, -4]\)	\(20346417/289\)	\(289\)	\([2, 2]\)	\(2\)	\(-0.72321\)
17.a1	17a3	\([1, -1, 1, -91, -310]\)	\(82483294977/17\)	\(17\)	\([2]\)	\(4\)	\(-0.37664\)
17.a4	17a4	\([1, -1, 1, -1, 0]\)	\(35937/17\)	\(17\)	\([4]\)	\(4\)	\(-1.0698\)