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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 60690.bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 60690.bt1 | 60690bu6 | \([1, 0, 0, -4855206, -4118151114]\) | \(524388516989299201/3150\) | \(76033342350\) | \([2]\) | \(1310720\) | \(2.1534\) | |
| 60690.bt2 | 60690bu4 | \([1, 0, 0, -303456, -64362564]\) | \(128031684631201/9922500\) | \(239505028402500\) | \([2, 2]\) | \(655360\) | \(1.8068\) | |
| 60690.bt3 | 60690bu5 | \([1, 0, 0, -283226, -73308270]\) | \(-104094944089921/35880468750\) | \(-866067290205468750\) | \([2]\) | \(1310720\) | \(2.1534\) | |
| 60690.bt4 | 60690bu3 | \([1, 0, 0, -106936, 12712580]\) | \(5602762882081/345888060\) | \(8348896914526140\) | \([2]\) | \(655360\) | \(1.8068\) | |
| 60690.bt5 | 60690bu2 | \([1, 0, 0, -20236, -864640]\) | \(37966934881/8643600\) | \(208635491408400\) | \([2, 2]\) | \(327680\) | \(1.4602\) | |
| 60690.bt6 | 60690bu1 | \([1, 0, 0, 2884, -83184]\) | \(109902239/188160\) | \(-4541724983040\) | \([2]\) | \(163840\) | \(1.1136\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 60690.bt have rank \(1\).
Complex multiplication
The elliptic curves in class 60690.bt do not have complex multiplication.Modular form 60690.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
