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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 127296cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127296.dn3 | 127296cc1 | \([0, 0, 0, -16104, 683192]\) | \(618724784128/87947613\) | \(65652541314048\) | \([2]\) | \(327680\) | \(1.3771\) | \(\Gamma_0(N)\)-optimal |
| 127296.dn2 | 127296cc2 | \([0, 0, 0, -68124, -6162640]\) | \(2927363579728/320445801\) | \(3827384138612736\) | \([2, 2]\) | \(655360\) | \(1.7237\) | |
| 127296.dn4 | 127296cc3 | \([0, 0, 0, 90996, -30667120]\) | \(1744147297148/9513325341\) | \(-454506196080328704\) | \([2]\) | \(1310720\) | \(2.0702\) | |
| 127296.dn1 | 127296cc4 | \([0, 0, 0, -1059564, -419791408]\) | \(2753580869496292/39328497\) | \(1878948204576768\) | \([2]\) | \(1310720\) | \(2.0702\) |
Rank
sage: E.rank()
The elliptic curves in class 127296cc have rank \(1\).
Complex multiplication
The elliptic curves in class 127296cc do not have complex multiplication.Modular form 127296.2.a.cc
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.
