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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 16562bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16562.bv5 | 16562bo1 | \([1, 1, 1, -4313, -222461]\) | \(-15625/28\) | \(-15900339057148\) | \([2]\) | \(34560\) | \(1.2233\) | \(\Gamma_0(N)\)-optimal |
| 16562.bv4 | 16562bo2 | \([1, 1, 1, -87123, -9927793]\) | \(128787625/98\) | \(55651186700018\) | \([2]\) | \(69120\) | \(1.5699\) | |
| 16562.bv6 | 16562bo3 | \([1, 1, 1, 37092, 4630205]\) | \(9938375/21952\) | \(-12465865820804032\) | \([2]\) | \(103680\) | \(1.7727\) | |
| 16562.bv3 | 16562bo4 | \([1, 1, 1, -294148, 50208829]\) | \(4956477625/941192\) | \(534473997066972872\) | \([2]\) | \(207360\) | \(2.1192\) | |
| 16562.bv2 | 16562bo5 | \([1, 1, 1, -1412083, 647136433]\) | \(-548347731625/1835008\) | \(-1042044620449251328\) | \([2]\) | \(311040\) | \(2.3220\) | |
| 16562.bv1 | 16562bo6 | \([1, 1, 1, -22611443, 41375346865]\) | \(2251439055699625/25088\) | \(14246703795204608\) | \([2]\) | \(622080\) | \(2.6685\) |
Rank
sage: E.rank()
The elliptic curves in class 16562bo have rank \(1\).
Complex multiplication
The elliptic curves in class 16562bo do not have complex multiplication.Modular form 16562.2.a.bo
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
