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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 182070.bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 182070.bw1 | 182070ct5 | \([1, -1, 0, -4532450634, 117449559112140]\) | \(585196747116290735872321/836876053125000\) | \(14725911884552465428125000\) | \([2]\) | \(169869312\) | \(4.1006\) | |
| 182070.bw2 | 182070ct4 | \([1, -1, 0, -657064674, -6481569972780]\) | \(1782900110862842086081/328139630024640\) | \(5774039368827226161416640\) | \([2]\) | \(84934656\) | \(3.7540\) | |
| 182070.bw3 | 182070ct3 | \([1, -1, 0, -285849954, 1800184153428]\) | \(146796951366228945601/5397929064360000\) | \(94983513345861211872360000\) | \([2, 2]\) | \(84934656\) | \(3.7540\) | |
| 182070.bw4 | 182070ct2 | \([1, -1, 0, -45309474, -79062400620]\) | \(584614687782041281/184812061593600\) | \(3252006224897124299673600\) | \([2, 2]\) | \(42467328\) | \(3.4074\) | |
| 182070.bw5 | 182070ct1 | \([1, -1, 0, 7959006, -8396435052]\) | \(3168685387909439/3563732336640\) | \(-62708459841247657328640\) | \([2]\) | \(21233664\) | \(3.0609\) | \(\Gamma_0(N)\)-optimal |
| 182070.bw6 | 182070ct6 | \([1, -1, 0, 112103046, 6419702168028]\) | \(8854313460877886399/1016927675429790600\) | \(-17894152049664511766764470600\) | \([2]\) | \(169869312\) | \(4.1006\) |
Rank
sage: E.rank()
The elliptic curves in class 182070.bw have rank \(0\).
Complex multiplication
The elliptic curves in class 182070.bw do not have complex multiplication.Modular form 182070.2.a.bw
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.
