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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 168.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
168.a1 | 168b4 | \([0, -1, 0, -4032, 99900]\) | \(7080974546692/189\) | \(193536\) | \([2]\) | \(96\) | \(0.52819\) | |
168.a2 | 168b3 | \([0, -1, 0, -392, -228]\) | \(6522128932/3720087\) | \(3809369088\) | \([2]\) | \(96\) | \(0.52819\) | |
168.a3 | 168b2 | \([0, -1, 0, -252, 1620]\) | \(6940769488/35721\) | \(9144576\) | \([2, 2]\) | \(48\) | \(0.18162\) | |
168.a4 | 168b1 | \([0, -1, 0, -7, 52]\) | \(-2725888/64827\) | \(-1037232\) | \([4]\) | \(24\) | \(-0.16496\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 168.a have rank \(0\).
Complex multiplication
The elliptic curves in class 168.a do not have complex multiplication.Modular form 168.2.a.a
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.