Properties

Label 13013a
Number of curves $3$
Conductor $13013$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 13013a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13013.j1 13013a1 \([0, 1, 1, -15097, 708972]\) \(-78843215872/539\) \(-2601650051\) \([]\) \(14400\) \(0.98748\) \(\Gamma_0(N)\)-optimal
13013.j2 13013a2 \([0, 1, 1, -8337, 1352017]\) \(-13278380032/156590819\) \(-755833974466571\) \([]\) \(43200\) \(1.5368\)  
13013.j3 13013a3 \([0, 1, 1, 74473, -35042978]\) \(9463555063808/115539436859\) \(-557686793685952931\) \([]\) \(129600\) \(2.0861\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13013a have rank \(2\).

Complex multiplication

The elliptic curves in class 13013a do not have complex multiplication.

Modular form 13013.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{4} - 3 q^{5} - q^{7} - 2 q^{9} + q^{11} - 2 q^{12} - 3 q^{15} + 4 q^{16} - 6 q^{17} - 2 q^{19} + O(q^{20})\) Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.