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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 24990by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 24990.ca5 | 24990by1 | \([1, 0, 0, 149939, -21693295]\) | \(3168685387909439/3563732336640\) | \(-419269545673359360\) | \([2]\) | \(442368\) | \(2.0679\) | \(\Gamma_0(N)\)-optimal |
| 24990.ca4 | 24990by2 | \([1, 0, 0, -853581, -204534639]\) | \(584614687782041281/184812061593600\) | \(21742954234425446400\) | \([2, 2]\) | \(884736\) | \(2.4145\) | |
| 24990.ca3 | 24990by3 | \([1, 0, 0, -5385101, 4654161105]\) | \(146796951366228945601/5397929064360000\) | \(635060956492889640000\) | \([2, 2]\) | \(1769472\) | \(2.7610\) | |
| 24990.ca2 | 24990by4 | \([1, 0, 0, -12378381, -16761062319]\) | \(1782900110862842086081/328139630024640\) | \(38605299332768871360\) | \([2]\) | \(1769472\) | \(2.7610\) | |
| 24990.ca6 | 24990by5 | \([1, 0, 0, 2111899, 16599880905]\) | \(8854313460877886399/1016927675429790600\) | \(-119640524086639434299400\) | \([2]\) | \(3538944\) | \(3.1076\) | |
| 24990.ca1 | 24990by6 | \([1, 0, 0, -85386421, 303683095001]\) | \(585196747116290735872321/836876053125000\) | \(98457630774103125000\) | \([2]\) | \(3538944\) | \(3.1076\) |
Rank
sage: E.rank()
The elliptic curves in class 24990by have rank \(0\).
Complex multiplication
The elliptic curves in class 24990by do not have complex multiplication.Modular form 24990.2.a.by
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
