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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 3330z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3330.v1 | 3330z1 | \([1, -1, 1, -482, -3949]\) | \(-16954786009/370\) | \(-269730\) | \([]\) | \(864\) | \(0.15809\) | \(\Gamma_0(N)\)-optimal |
3330.v2 | 3330z2 | \([1, -1, 1, -167, -9241]\) | \(-702595369/50653000\) | \(-36926037000\) | \([3]\) | \(2592\) | \(0.70740\) | |
3330.v3 | 3330z3 | \([1, -1, 1, 1498, 248501]\) | \(510273943271/37000000000\) | \(-26973000000000\) | \([3]\) | \(7776\) | \(1.2567\) |
Rank
sage: E.rank()
The elliptic curves in class 3330z have rank \(1\).
Complex multiplication
The elliptic curves in class 3330z do not have complex multiplication.Modular form 3330.2.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 Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.