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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 453152.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
453152.u1 | 453152u4 | \([0, 0, 0, -155771, -23592226]\) | \(287496\) | \(1453957557903872\) | \([2]\) | \(1966080\) | \(1.7722\) | \(-16\) | |
453152.u2 | 453152u2 | \([0, 0, 0, -155771, 23592226]\) | \(287496\) | \(1453957557903872\) | \([2]\) | \(1966080\) | \(1.7722\) | \(\Gamma_0(N)\)-optimal* | \(-16\) |
453152.u3 | 453152u1 | \([0, 0, 0, -14161, 0]\) | \(1728\) | \(181744694737984\) | \([2, 2]\) | \(983040\) | \(1.4256\) | \(\Gamma_0(N)\)-optimal* | \(-4\) |
453152.u4 | 453152u3 | \([0, 0, 0, 56644, 0]\) | \(1728\) | \(-11631660463230976\) | \([2]\) | \(1966080\) | \(1.7722\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 453152.u have rank \(1\).
Complex multiplication
Each elliptic curve in class 453152.u has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 453152.2.a.u
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.