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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 137280cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 137280.dr3 | 137280cr1 | \([0, -1, 0, -4225, 77857]\) | \(31824875809/8785920\) | \(2303176212480\) | \([2]\) | \(221184\) | \(1.0796\) | \(\Gamma_0(N)\)-optimal |
| 137280.dr2 | 137280cr2 | \([0, -1, 0, -24705, -1425375]\) | \(6361447449889/294465600\) | \(77192390246400\) | \([2, 2]\) | \(442368\) | \(1.4262\) | |
| 137280.dr4 | 137280cr3 | \([0, -1, 0, 13695, -5488095]\) | \(1083523132511/50179392120\) | \(-13154226567905280\) | \([4]\) | \(884736\) | \(1.7727\) | |
| 137280.dr1 | 137280cr4 | \([0, -1, 0, -390785, -93897183]\) | \(25176685646263969/57915000\) | \(15182069760000\) | \([2]\) | \(884736\) | \(1.7727\) |
Rank
sage: E.rank()
The elliptic curves in class 137280cr have rank \(1\).
Complex multiplication
The elliptic curves in class 137280cr do not have complex multiplication.Modular form 137280.2.a.cr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
