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SageMath
E = EllipticCurve("fx1")
E.isogeny_class()
Elliptic curves in class 28224fx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28224.eq3 | 28224fx1 | \([0, 0, 0, -5439, 128968]\) | \(3241792/567\) | \(3112280998848\) | \([2]\) | \(49152\) | \(1.1171\) | \(\Gamma_0(N)\)-optimal |
28224.eq2 | 28224fx2 | \([0, 0, 0, -25284, -1426880]\) | \(5088448/441\) | \(154922431942656\) | \([2, 2]\) | \(98304\) | \(1.4636\) | |
28224.eq4 | 28224fx3 | \([0, 0, 0, 27636, -6613040]\) | \(830584/7203\) | \(-20243197773840384\) | \([2]\) | \(196608\) | \(1.8102\) | |
28224.eq1 | 28224fx4 | \([0, 0, 0, -395724, -95814992]\) | \(2438569736/21\) | \(59018069311488\) | \([2]\) | \(196608\) | \(1.8102\) |
Rank
sage: E.rank()
The elliptic curves in class 28224fx have rank \(0\).
Complex multiplication
The elliptic curves in class 28224fx do not have complex multiplication.Modular form 28224.2.a.fx
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.