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SageMath
E = EllipticCurve("dn1")
E.isogeny_class()
Elliptic curves in class 28224.dn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 28224.dn1 | 28224dj2 | \([0, 0, 0, 0, -6048]\) | \(0\) | \(-15801827328\) | \([]\) | \(13824\) | \(0.63583\) | \(-3\) | |
| 28224.dn2 | 28224dj1 | \([0, 0, 0, 0, 224]\) | \(0\) | \(-21676032\) | \([]\) | \(4608\) | \(0.086526\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 28224.dn have rank \(1\).
Complex multiplication
Each elliptic curve in class 28224.dn has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 28224.2.a.dn
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
