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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)
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.