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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 382200u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
382200.u3 | 382200u1 | \([0, -1, 0, -557783, 160527312]\) | \(652517349376/1365\) | \(40147721250000\) | \([4]\) | \(2949120\) | \(1.8593\) | \(\Gamma_0(N)\)-optimal |
382200.u2 | 382200u2 | \([0, -1, 0, -563908, 156827812]\) | \(42140629456/1863225\) | \(876826232100000000\) | \([2, 2]\) | \(5898240\) | \(2.2059\) | |
382200.u4 | 382200u3 | \([0, -1, 0, 293592, 590722812]\) | \(1486779836/80970435\) | \(-152417451317040000000\) | \([2]\) | \(11796480\) | \(2.5524\) | |
382200.u1 | 382200u4 | \([0, -1, 0, -1519408, -513933188]\) | \(206081497444/58524375\) | \(110165347110000000000\) | \([2]\) | \(11796480\) | \(2.5524\) |
Rank
sage: E.rank()
The elliptic curves in class 382200u have rank \(2\).
Complex multiplication
The elliptic curves in class 382200u do not have complex multiplication.Modular form 382200.2.a.u
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.