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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 3344.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3344.f1 | 3344a3 | \([0, 0, 0, -851, -9486]\) | \(33279932754/278179\) | \(569710592\) | \([2]\) | \(1152\) | \(0.50577\) | |
3344.f2 | 3344a2 | \([0, 0, 0, -91, 90]\) | \(81385668/43681\) | \(44729344\) | \([2, 2]\) | \(576\) | \(0.15919\) | |
3344.f3 | 3344a1 | \([0, 0, 0, -71, 230]\) | \(154617552/209\) | \(53504\) | \([2]\) | \(288\) | \(-0.18738\) | \(\Gamma_0(N)\)-optimal |
3344.f4 | 3344a4 | \([0, 0, 0, 349, 706]\) | \(2295461646/1433531\) | \(-2935871488\) | \([2]\) | \(1152\) | \(0.50577\) |
Rank
sage: E.rank()
The elliptic curves in class 3344.f have rank \(0\).
Complex multiplication
The elliptic curves in class 3344.f do not have complex multiplication.Modular form 3344.2.a.f
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 LMFDB 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 LMFDB labels.