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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 84672er
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84672.kh2 | 84672er1 | \([0, 0, 0, -9996, -389648]\) | \(-132651/2\) | \(-1665412890624\) | \([]\) | \(145152\) | \(1.1482\) | \(\Gamma_0(N)\)-optimal |
| 84672.kh3 | 84672er2 | \([0, 0, 0, 37044, -1926288]\) | \(9261/8\) | \(-4856343989059584\) | \([]\) | \(435456\) | \(1.6975\) | |
| 84672.kh1 | 84672er3 | \([0, 0, 0, -386316, 122541552]\) | \(-1167051/512\) | \(-2797254137698320384\) | \([]\) | \(1306368\) | \(2.2468\) |
Rank
sage: E.rank()
The elliptic curves in class 84672er have rank \(1\).
Complex multiplication
The elliptic curves in class 84672er do not have complex multiplication.Modular form 84672.2.a.er
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.
