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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 300033f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
300033.f1 | 300033f1 | \([1, -1, 1, -6224, -187414]\) | \(36571225840057/2700297\) | \(1968516513\) | \([2]\) | \(362496\) | \(0.83360\) | \(\Gamma_0(N)\)-optimal |
300033.f2 | 300033f2 | \([1, -1, 1, -5819, -213172]\) | \(-29886240312937/10002200121\) | \(-7291603888209\) | \([2]\) | \(724992\) | \(1.1802\) |
Rank
sage: E.rank()
The elliptic curves in class 300033f have rank \(1\).
Complex multiplication
The elliptic curves in class 300033f do not have complex multiplication.Modular form 300033.2.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 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.