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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 3192p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3192.l2 | 3192p1 | \([0, 1, 0, -180, 864]\) | \(2533446736/25137\) | \(6435072\) | \([2]\) | \(1152\) | \(0.12521\) | \(\Gamma_0(N)\)-optimal |
| 3192.l1 | 3192p2 | \([0, 1, 0, -320, -816]\) | \(3550014724/1842183\) | \(1886395392\) | \([2]\) | \(2304\) | \(0.47179\) |
Rank
sage: E.rank()
The elliptic curves in class 3192p have rank \(1\).
Complex multiplication
The elliptic curves in class 3192p do not have complex multiplication.Modular form 3192.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
