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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 18810u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18810.t4 | 18810u1 | \([1, -1, 1, -103538, -61309583]\) | \(-168380411424176601/2131914391552000\) | \(-1554165591441408000\) | \([2]\) | \(294912\) | \(2.1724\) | \(\Gamma_0(N)\)-optimal |
18810.t3 | 18810u2 | \([1, -1, 1, -3052658, -2045477519]\) | \(4315493878427398863321/16147293184000000\) | \(11771376731136000000\) | \([2, 2]\) | \(589824\) | \(2.5189\) | |
18810.t1 | 18810u3 | \([1, -1, 1, -48798578, -131195358863]\) | \(17628594000102642361428441/248187500000000\) | \(180928687500000000\) | \([2]\) | \(1179648\) | \(2.8655\) | |
18810.t2 | 18810u4 | \([1, -1, 1, -4492658, 82266481]\) | \(13756443594716753103321/7957003087464992000\) | \(5800655250761979168000\) | \([2]\) | \(1179648\) | \(2.8655\) |
Rank
sage: E.rank()
The elliptic curves in class 18810u have rank \(0\).
Complex multiplication
The elliptic curves in class 18810u do not have complex multiplication.Modular form 18810.2.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.