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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 5408a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
5408.g3 | 5408a1 | \([0, 0, 0, -169, 0]\) | \(1728\) | \(308915776\) | \([2, 2]\) | \(960\) | \(0.31852\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
5408.g1 | 5408a2 | \([0, 0, 0, -1859, -30758]\) | \(287496\) | \(2471326208\) | \([2]\) | \(1920\) | \(0.66509\) | \(-16\) | |
5408.g2 | 5408a3 | \([0, 0, 0, -1859, 30758]\) | \(287496\) | \(2471326208\) | \([2]\) | \(1920\) | \(0.66509\) | \(-16\) | |
5408.g4 | 5408a4 | \([0, 0, 0, 676, 0]\) | \(1728\) | \(-19770609664\) | \([2]\) | \(1920\) | \(0.66509\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 5408a have rank \(1\).
Complex multiplication
Each elliptic curve in class 5408a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 5408.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.