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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 3366p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3366.n1 | 3366p1 | \([1, -1, 1, -290, 289]\) | \(3687953625/2106368\) | \(1535542272\) | \([2]\) | \(1280\) | \(0.45244\) | \(\Gamma_0(N)\)-optimal |
3366.n2 | 3366p2 | \([1, -1, 1, 1150, 1441]\) | \(230910510375/135399968\) | \(-98706576672\) | \([2]\) | \(2560\) | \(0.79902\) |
Rank
sage: E.rank()
The elliptic curves in class 3366p have rank \(1\).
Complex multiplication
The elliptic curves in class 3366p do not have complex multiplication.Modular form 3366.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.