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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 8112.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8112.m1 | 8112u1 | \([0, -1, 0, -167704, -46227728]\) | \(-156116857/186624\) | \(-623554273590902784\) | \([]\) | \(119808\) | \(2.1083\) | \(\Gamma_0(N)\)-optimal |
| 8112.m2 | 8112u2 | \([0, -1, 0, 1414136, 864912112]\) | \(93603087383/150994944\) | \(-504509294741400059904\) | \([]\) | \(359424\) | \(2.6577\) |
Rank
sage: E.rank()
The elliptic curves in class 8112.m have rank \(0\).
Complex multiplication
The elliptic curves in class 8112.m do not have complex multiplication.Modular form 8112.2.a.m
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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
