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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 75106b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75106.a4 | 75106b1 | \([1, 0, 0, -6673, -130455]\) | \(3048625/1088\) | \(11727786277952\) | \([2]\) | \(211968\) | \(1.2087\) | \(\Gamma_0(N)\)-optimal |
| 75106.a3 | 75106b2 | \([1, 0, 0, -95033, -11281487]\) | \(8805624625/2312\) | \(24921545840648\) | \([2]\) | \(423936\) | \(1.5553\) | |
| 75106.a2 | 75106b3 | \([1, 0, 0, -227573, 41761021]\) | \(120920208625/19652\) | \(211833139645508\) | \([2]\) | \(635904\) | \(1.7580\) | |
| 75106.a1 | 75106b4 | \([1, 0, 0, -249663, 33159175]\) | \(159661140625/48275138\) | \(520368107539190402\) | \([2]\) | \(1271808\) | \(2.1046\) |
Rank
sage: E.rank()
The elliptic curves in class 75106b have rank \(1\).
Complex multiplication
The elliptic curves in class 75106b do not have complex multiplication.Modular form 75106.2.a.b
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.
