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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 3192.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3192.a1 | 3192m1 | \([0, -1, 0, -106435, -13329764]\) | \(8334147900493981696/232793757\) | \(3724700112\) | \([2]\) | \(11520\) | \(1.3474\) | \(\Gamma_0(N)\)-optimal |
| 3192.a2 | 3192m2 | \([0, -1, 0, -106300, -13365404]\) | \(-518904725785387216/2753286252003\) | \(-704841280512768\) | \([2]\) | \(23040\) | \(1.6940\) |
Rank
sage: E.rank()
The elliptic curves in class 3192.a have rank \(1\).
Complex multiplication
The elliptic curves in class 3192.a do not have complex multiplication.Modular form 3192.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
