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SageMath
E = EllipticCurve("db1")
E.isogeny_class()
Elliptic curves in class 141120db
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
141120.z5 | 141120db1 | \([0, 0, 0, -27048, -7043848]\) | \(-24918016/229635\) | \(-20167580872535040\) | \([2]\) | \(786432\) | \(1.8110\) | \(\Gamma_0(N)\)-optimal |
141120.z4 | 141120db2 | \([0, 0, 0, -741468, -245088592]\) | \(32082281296/99225\) | \(139430188748390400\) | \([2, 2]\) | \(1572864\) | \(2.1576\) | |
141120.z3 | 141120db3 | \([0, 0, 0, -1058988, -14823088]\) | \(23366901604/13505625\) | \(75911991651901440000\) | \([2, 2]\) | \(3145728\) | \(2.5042\) | |
141120.z1 | 141120db4 | \([0, 0, 0, -11854668, -15710217712]\) | \(32779037733124/315\) | \(1770542079344640\) | \([2]\) | \(3145728\) | \(2.5042\) | |
141120.z2 | 141120db5 | \([0, 0, 0, -11431308, 14825892368]\) | \(14695548366242/57421875\) | \(645510133094400000000\) | \([2]\) | \(6291456\) | \(2.8507\) | |
141120.z6 | 141120db6 | \([0, 0, 0, 4233012, -118546288]\) | \(746185003198/432360075\) | \(-4860391785499076198400\) | \([2]\) | \(6291456\) | \(2.8507\) |
Rank
sage: E.rank()
The elliptic curves in class 141120db have rank \(0\).
Complex multiplication
The elliptic curves in class 141120db do not have complex multiplication.Modular form 141120.2.a.db
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.