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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 101640.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 101640.bf1 | 101640p6 | \([0, -1, 0, -1268120, 550063692]\) | \(62161150998242/1607445\) | \(5832062713128960\) | \([2]\) | \(1310720\) | \(2.1319\) | |
| 101640.bf2 | 101640p4 | \([0, -1, 0, -82320, 7915932]\) | \(34008619684/4862025\) | \(8820094843929600\) | \([2, 2]\) | \(655360\) | \(1.7853\) | |
| 101640.bf3 | 101640p2 | \([0, -1, 0, -21820, -1110668]\) | \(2533446736/275625\) | \(125001344160000\) | \([2, 2]\) | \(327680\) | \(1.4387\) | |
| 101640.bf4 | 101640p1 | \([0, -1, 0, -21215, -1182300]\) | \(37256083456/525\) | \(14881112400\) | \([2]\) | \(163840\) | \(1.0921\) | \(\Gamma_0(N)\)-optimal |
| 101640.bf5 | 101640p3 | \([0, -1, 0, 29000, -5562500]\) | \(1486779836/8203125\) | \(-14881112400000000\) | \([2]\) | \(655360\) | \(1.7853\) | |
| 101640.bf6 | 101640p5 | \([0, -1, 0, 135480, 42502572]\) | \(75798394558/259416045\) | \(-941202120901109760\) | \([2]\) | \(1310720\) | \(2.1319\) |
Rank
sage: E.rank()
The elliptic curves in class 101640.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 101640.bf do not have complex multiplication.Modular form 101640.2.a.bf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
