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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 167310bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 167310.ei3 | 167310bt1 | \([1, -1, 1, -100418, 8882241]\) | \(31824875809/8785920\) | \(30915401184645120\) | \([2]\) | \(1548288\) | \(1.8716\) | \(\Gamma_0(N)\)-optimal |
| 167310.ei2 | 167310bt2 | \([1, -1, 1, -587138, -165947583]\) | \(6361447449889/294465600\) | \(1036148992829121600\) | \([2, 2]\) | \(3096576\) | \(2.2182\) | |
| 167310.ei4 | 167310bt3 | \([1, -1, 1, 325462, -635389023]\) | \(1083523132511/50179392120\) | \(-176568422953022563320\) | \([2]\) | \(6193152\) | \(2.5648\) | |
| 167310.ei1 | 167310bt4 | \([1, -1, 1, -9287258, -10891455519]\) | \(25176685646263969/57915000\) | \(203788044918315000\) | \([2]\) | \(6193152\) | \(2.5648\) |
Rank
sage: E.rank()
The elliptic curves in class 167310bt have rank \(0\).
Complex multiplication
The elliptic curves in class 167310bt do not have complex multiplication.Modular form 167310.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
