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SageMath
E = EllipticCurve("fq1")
E.isogeny_class()
Elliptic curves in class 114240.fq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 114240.fq1 | 114240dk6 | \([0, 1, 0, -111525121, -453359005345]\) | \(585196747116290735872321/836876053125000\) | \(219382036070400000000\) | \([2]\) | \(14155776\) | \(3.1744\) | |
| 114240.fq2 | 114240dk4 | \([0, 1, 0, -16167681, 25012499295]\) | \(1782900110862842086081/328139630024640\) | \(86019835173179228160\) | \([2]\) | \(7077888\) | \(2.8278\) | |
| 114240.fq3 | 114240dk3 | \([0, 1, 0, -7033601, -6950333601]\) | \(146796951366228945601/5397929064360000\) | \(1415034716647587840000\) | \([2, 2]\) | \(7077888\) | \(2.8278\) | |
| 114240.fq4 | 114240dk2 | \([0, 1, 0, -1114881, 304833375]\) | \(584614687782041281/184812061593600\) | \(48447373074392678400\) | \([2, 2]\) | \(3538944\) | \(2.4812\) | |
| 114240.fq5 | 114240dk1 | \([0, 1, 0, 195839, 32465759]\) | \(3168685387909439/3563732336640\) | \(-934211049656156160\) | \([2]\) | \(1769472\) | \(2.1347\) | \(\Gamma_0(N)\)-optimal |
| 114240.fq6 | 114240dk5 | \([0, 1, 0, 2758399, -24777648801]\) | \(8854313460877886399/1016927675429790600\) | \(-266581488547867027046400\) | \([2]\) | \(14155776\) | \(3.1744\) |
Rank
sage: E.rank()
The elliptic curves in class 114240.fq have rank \(1\).
Complex multiplication
The elliptic curves in class 114240.fq do not have complex multiplication.Modular form 114240.2.a.fq
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
