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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 164730cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 164730.v4 | 164730cr1 | \([1, 1, 0, -412842, 129802836]\) | \(-322391399464009/115780151040\) | \(-2794651384558421760\) | \([2]\) | \(3538944\) | \(2.2502\) | \(\Gamma_0(N)\)-optimal |
| 164730.v3 | 164730cr2 | \([1, 1, 0, -7094522, 7269846084]\) | \(1636061778667305289/135585968400\) | \(3272715667686819600\) | \([2, 2]\) | \(7077888\) | \(2.5968\) | |
| 164730.v1 | 164730cr3 | \([1, 1, 0, -113510102, 465431484216]\) | \(6700909177116065071369/46027500\) | \(1110991957147500\) | \([4]\) | \(14155776\) | \(2.9434\) | |
| 164730.v2 | 164730cr4 | \([1, 1, 0, -7585822, 6204609424]\) | \(2000037860254622089/467727326149140\) | \(11289800608110371040660\) | \([2]\) | \(14155776\) | \(2.9434\) |
Rank
sage: E.rank()
The elliptic curves in class 164730cr have rank \(0\).
Complex multiplication
The elliptic curves in class 164730cr do not have complex multiplication.Modular form 164730.2.a.cr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
