Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 17a

sage: E.isogeny_class().curves
 
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 \([1, -1, 1, -1, 0]\) \(35937/17\) \(17\) \([4]\) \(4\) \(-1.0698\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 17.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - 2q^{5} + 4q^{7} + 3q^{8} - 3q^{9} + 2q^{10} - 2q^{13} - 4q^{14} - q^{16} + q^{17} + 3q^{18} - 4q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().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

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.