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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 4032o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4032.bh4 | 4032o1 | \([0, 0, 0, -264, -10712]\) | \(-2725888/64827\) | \(-48393096192\) | \([2]\) | \(3072\) | \(0.73092\) | \(\Gamma_0(N)\)-optimal |
| 4032.bh3 | 4032o2 | \([0, 0, 0, -9084, -331760]\) | \(6940769488/35721\) | \(426649337856\) | \([2, 2]\) | \(6144\) | \(1.0775\) | |
| 4032.bh1 | 4032o3 | \([0, 0, 0, -145164, -21288080]\) | \(7080974546692/189\) | \(9029615616\) | \([2]\) | \(12288\) | \(1.4241\) | |
| 4032.bh2 | 4032o4 | \([0, 0, 0, -14124, 77488]\) | \(6522128932/3720087\) | \(177729924169728\) | \([2]\) | \(12288\) | \(1.4241\) |
Rank
sage: E.rank()
The elliptic curves in class 4032o have rank \(1\).
Complex multiplication
The elliptic curves in class 4032o do not have complex multiplication.Modular form 4032.2.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.
