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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 68544cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68544.bg3 | 68544cc1 | \([0, 0, 0, -4296, -108376]\) | \(11745974272/357\) | \(266499072\) | \([2]\) | \(40960\) | \(0.71461\) | \(\Gamma_0(N)\)-optimal |
| 68544.bg2 | 68544cc2 | \([0, 0, 0, -4476, -98800]\) | \(830321872/127449\) | \(1522242699264\) | \([2, 2]\) | \(81920\) | \(1.0612\) | |
| 68544.bg4 | 68544cc3 | \([0, 0, 0, 7764, -544336]\) | \(1083360092/3306177\) | \(-157955065970688\) | \([2]\) | \(163840\) | \(1.4078\) | |
| 68544.bg1 | 68544cc4 | \([0, 0, 0, -19596, 959600]\) | \(17418812548/1753941\) | \(83795836207104\) | \([2]\) | \(163840\) | \(1.4078\) |
Rank
sage: E.rank()
The elliptic curves in class 68544cc have rank \(1\).
Complex multiplication
The elliptic curves in class 68544cc do not have complex multiplication.Modular form 68544.2.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.
