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SageMath
E = EllipticCurve("db1")
E.isogeny_class()
Elliptic curves in class 85680.db
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85680.db1 | 85680el6 | \([0, 0, 0, -250931523, -1529961177278]\) | \(585196747116290735872321/836876053125000\) | \(2498898504614400000000\) | \([2]\) | \(14155776\) | \(3.3771\) | |
| 85680.db2 | 85680el4 | \([0, 0, 0, -36377283, 84435373762]\) | \(1782900110862842086081/328139630024640\) | \(979819685019494645760\) | \([2]\) | \(7077888\) | \(3.0305\) | |
| 85680.db3 | 85680el3 | \([0, 0, 0, -15825603, -23449463102]\) | \(146796951366228945601/5397929064360000\) | \(16118129819313930240000\) | \([2, 2]\) | \(7077888\) | \(3.0305\) | |
| 85680.db4 | 85680el2 | \([0, 0, 0, -2508483, 1030066882]\) | \(584614687782041281/184812061593600\) | \(551845858925504102400\) | \([2, 2]\) | \(3538944\) | \(2.6840\) | |
| 85680.db5 | 85680el1 | \([0, 0, 0, 440637, 109351618]\) | \(3168685387909439/3563732336640\) | \(-10641247737489653760\) | \([2]\) | \(1769472\) | \(2.3374\) | \(\Gamma_0(N)\)-optimal |
| 85680.db6 | 85680el5 | \([0, 0, 0, 6206397, -83627667902]\) | \(8854313460877886399/1016927675429790600\) | \(-3036529767990547854950400\) | \([2]\) | \(14155776\) | \(3.3771\) |
Rank
sage: E.rank()
The elliptic curves in class 85680.db have rank \(0\).
Complex multiplication
The elliptic curves in class 85680.db do not have complex multiplication.Modular form 85680.2.a.db
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.
