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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 160080bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 160080.h4 | 160080bt1 | \([0, -1, 0, 67984, -8184384]\) | \(8483547917294351/11935126180800\) | \(-48886276836556800\) | \([2]\) | \(1105920\) | \(1.8880\) | \(\Gamma_0(N)\)-optimal |
| 160080.h3 | 160080bt2 | \([0, -1, 0, -433136, -80746560]\) | \(2194004991989474929/579560505435000\) | \(2373879830261760000\) | \([2]\) | \(2211840\) | \(2.2345\) | |
| 160080.h2 | 160080bt3 | \([0, -1, 0, -660656, 340344000]\) | \(-7785572839582076209/7695526715437500\) | \(-31520877426432000000\) | \([2]\) | \(3317760\) | \(2.4373\) | |
| 160080.h1 | 160080bt4 | \([0, -1, 0, -12367376, 16739117376]\) | \(51073602635162593292689/18899093261718750\) | \(77410686000000000000\) | \([2]\) | \(6635520\) | \(2.7839\) |
Rank
sage: E.rank()
The elliptic curves in class 160080bt have rank \(0\).
Complex multiplication
The elliptic curves in class 160080bt do not have complex multiplication.Modular form 160080.2.a.bt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
