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SageMath
E = EllipticCurve("di1")
E.isogeny_class()
Elliptic curves in class 28224di
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
28224.di4 | 28224di1 | \([0, 0, 0, 0, 2744]\) | \(0\) | \(-3252759552\) | \([2]\) | \(9216\) | \(0.50411\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
28224.di2 | 28224di2 | \([0, 0, 0, -2940, 60368]\) | \(54000\) | \(52044152832\) | \([2]\) | \(18432\) | \(0.85069\) | \(-12\) | |
28224.di3 | 28224di3 | \([0, 0, 0, 0, -74088]\) | \(0\) | \(-2371261713408\) | \([2]\) | \(27648\) | \(1.0534\) | \(-3\) | |
28224.di1 | 28224di4 | \([0, 0, 0, -26460, -1629936]\) | \(54000\) | \(37940187414528\) | \([2]\) | \(55296\) | \(1.4000\) | \(-12\) |
Rank
sage: E.rank()
The elliptic curves in class 28224di have rank \(1\).
Complex multiplication
Each elliptic curve in class 28224di has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 28224.2.a.di
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.