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SageMath
E = EllipticCurve("fi1")
E.isogeny_class()
Elliptic curves in class 100800fi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.pa5 | 100800fi1 | \([0, 0, 0, -13800, -2567000]\) | \(-24918016/229635\) | \(-2678462640000000\) | \([2]\) | \(393216\) | \(1.6428\) | \(\Gamma_0(N)\)-optimal |
| 100800.pa4 | 100800fi2 | \([0, 0, 0, -378300, -89318000]\) | \(32082281296/99225\) | \(18517766400000000\) | \([2, 2]\) | \(786432\) | \(1.9894\) | |
| 100800.pa3 | 100800fi3 | \([0, 0, 0, -540300, -5402000]\) | \(23366901604/13505625\) | \(10081895040000000000\) | \([2, 2]\) | \(1572864\) | \(2.3359\) | |
| 100800.pa1 | 100800fi4 | \([0, 0, 0, -6048300, -5725298000]\) | \(32779037733124/315\) | \(235146240000000\) | \([2]\) | \(1572864\) | \(2.3359\) | |
| 100800.pa6 | 100800fi5 | \([0, 0, 0, 2159700, -43202000]\) | \(746185003198/432360075\) | \(-645510133094400000000\) | \([2]\) | \(3145728\) | \(2.6825\) | |
| 100800.pa2 | 100800fi6 | \([0, 0, 0, -5832300, 5403022000]\) | \(14695548366242/57421875\) | \(85730400000000000000\) | \([2]\) | \(3145728\) | \(2.6825\) |
Rank
sage: E.rank()
The elliptic curves in class 100800fi have rank \(1\).
Complex multiplication
The elliptic curves in class 100800fi do not have complex multiplication.Modular form 100800.2.a.fi
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.
