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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 27200z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27200.p4 | 27200z1 | \([0, 1, 0, -4833, 78463]\) | \(3048625/1088\) | \(4456448000000\) | \([2]\) | \(55296\) | \(1.1280\) | \(\Gamma_0(N)\)-optimal |
| 27200.p3 | 27200z2 | \([0, 1, 0, -68833, 6926463]\) | \(8805624625/2312\) | \(9469952000000\) | \([2]\) | \(110592\) | \(1.4746\) | |
| 27200.p2 | 27200z3 | \([0, 1, 0, -164833, -25809537]\) | \(120920208625/19652\) | \(80494592000000\) | \([2]\) | \(165888\) | \(1.6774\) | |
| 27200.p1 | 27200z4 | \([0, 1, 0, -180833, -20513537]\) | \(159661140625/48275138\) | \(197734965248000000\) | \([2]\) | \(331776\) | \(2.0239\) |
Rank
sage: E.rank()
The elliptic curves in class 27200z have rank \(0\).
Complex multiplication
The elliptic curves in class 27200z do not have complex multiplication.Modular form 27200.2.a.z
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.
