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SageMath
E = EllipticCurve("fa1")
E.isogeny_class()
Elliptic curves in class 28224.fa
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
28224.fa1 | 28224ds2 | \([0, 0, 0, -84, 0]\) | \(1728\) | \(37933056\) | \([2]\) | \(4096\) | \(0.14374\) | \(-4\) | |
28224.fa2 | 28224ds1 | \([0, 0, 0, 21, 0]\) | \(1728\) | \(-592704\) | \([2]\) | \(2048\) | \(-0.20283\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 28224.fa have rank \(1\).
Complex multiplication
Each elliptic curve in class 28224.fa has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 28224.2.a.fa
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.