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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 134064.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
134064.z1 | 134064ds4 | \([0, 0, 0, -261435531, -1627027786006]\) | \(22501000029889239268/3620708343\) | \(317986928487882243072\) | \([2]\) | \(18874368\) | \(3.3364\) | |
134064.z2 | 134064ds2 | \([0, 0, 0, -16389471, -25259710210]\) | \(22174957026242512/278654127129\) | \(6118165397630770331904\) | \([2, 2]\) | \(9437184\) | \(2.9898\) | |
134064.z3 | 134064ds3 | \([0, 0, 0, -2815491, -65832336430]\) | \(-28104147578308/21301741002339\) | \(-1870814921028882387987456\) | \([2]\) | \(18874368\) | \(3.3364\) | |
134064.z4 | 134064ds1 | \([0, 0, 0, -1922466, 401863259]\) | \(572616640141312/280535480757\) | \(384967039798368537552\) | \([2]\) | \(4718592\) | \(2.6433\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 134064.z have rank \(0\).
Complex multiplication
The elliptic curves in class 134064.z do not have complex multiplication.Modular form 134064.2.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 LMFDB 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 LMFDB labels.