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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 10080.bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10080.bt1 | 10080cf2 | \([0, 0, 0, -1692, -26784]\) | \(179406144/35\) | \(104509440\) | \([2]\) | \(6144\) | \(0.53847\) | |
| 10080.bt2 | 10080cf3 | \([0, 0, 0, -747, 7614]\) | \(123505992/4375\) | \(1632960000\) | \([2]\) | \(6144\) | \(0.53847\) | |
| 10080.bt3 | 10080cf1 | \([0, 0, 0, -117, -324]\) | \(3796416/1225\) | \(57153600\) | \([2, 2]\) | \(3072\) | \(0.19190\) | \(\Gamma_0(N)\)-optimal |
| 10080.bt4 | 10080cf4 | \([0, 0, 0, 333, -2214]\) | \(10941048/12005\) | \(-4480842240\) | \([2]\) | \(6144\) | \(0.53847\) |
Rank
sage: E.rank()
The elliptic curves in class 10080.bt have rank \(1\).
Complex multiplication
The elliptic curves in class 10080.bt do not have complex multiplication.Modular form 10080.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
