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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 32946.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32946.r1 | 32946r3 | \([1, 1, 1, -123698, -16796677]\) | \(8671983378625/82308\) | \(1986715029252\) | \([2]\) | \(165888\) | \(1.5221\) | |
| 32946.r2 | 32946r4 | \([1, 1, 1, -120808, -17615125]\) | \(-8078253774625/846825858\) | \(-20440317578459202\) | \([2]\) | \(331776\) | \(1.8687\) | |
| 32946.r3 | 32946r1 | \([1, 1, 1, -2318, 2315]\) | \(57066625/32832\) | \(792484665408\) | \([2]\) | \(55296\) | \(0.97280\) | \(\Gamma_0(N)\)-optimal |
| 32946.r4 | 32946r2 | \([1, 1, 1, 9242, 30059]\) | \(3616805375/2105352\) | \(-50818079169288\) | \([2]\) | \(110592\) | \(1.3194\) |
Rank
sage: E.rank()
The elliptic curves in class 32946.r have rank \(1\).
Complex multiplication
The elliptic curves in class 32946.r do not have complex multiplication.Modular form 32946.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
