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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 4368o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4368.k3 | 4368o1 | \([0, -1, 0, -112, 448]\) | \(38272753/4368\) | \(17891328\) | \([2]\) | \(1152\) | \(0.12397\) | \(\Gamma_0(N)\)-optimal |
| 4368.k2 | 4368o2 | \([0, -1, 0, -432, -2880]\) | \(2181825073/298116\) | \(1221083136\) | \([2, 2]\) | \(2304\) | \(0.47055\) | |
| 4368.k1 | 4368o3 | \([0, -1, 0, -6672, -207552]\) | \(8020417344913/187278\) | \(767090688\) | \([2]\) | \(4608\) | \(0.81712\) | |
| 4368.k4 | 4368o4 | \([0, -1, 0, 688, -16320]\) | \(8780064047/32388174\) | \(-132661960704\) | \([2]\) | \(4608\) | \(0.81712\) |
Rank
sage: E.rank()
The elliptic curves in class 4368o have rank \(0\).
Complex multiplication
The elliptic curves in class 4368o do not have complex multiplication.Modular form 4368.2.a.o
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.
