Properties

Label 73997j
Number of curves $3$
Conductor $73997$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 73997j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
73997.c1 73997j1 \([0, -1, 1, -85849, -9653183]\) \(-78843215872/539\) \(-478364484059\) \([]\) \(199800\) \(1.4220\) \(\Gamma_0(N)\)-optimal
73997.c2 73997j2 \([0, -1, 1, -47409, -18355038]\) \(-13278380032/156590819\) \(-138974928273304739\) \([]\) \(599400\) \(1.9713\)  
73997.c3 73997j3 \([0, -1, 1, 423481, 475373127]\) \(9463555063808/115539436859\) \(-102541675513029577979\) \([]\) \(1798200\) \(2.5206\)  

Rank

sage: E.rank()
 

The elliptic curves in class 73997j have rank \(0\).

Complex multiplication

The elliptic curves in class 73997j do not have complex multiplication.

Modular form 73997.2.a.j

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{4} + 3 q^{5} + q^{7} - 2 q^{9} + q^{11} + 2 q^{12} + 4 q^{13} - 3 q^{15} + 4 q^{16} + 6 q^{17} + 2 q^{19} + O(q^{20})\) 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.