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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 47190i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47190.o2 | 47190i1 | \([1, 1, 0, 163, -1239]\) | \(356400829/760500\) | \(-1012225500\) | \([2]\) | \(20736\) | \(0.41222\) | \(\Gamma_0(N)\)-optimal |
47190.o1 | 47190i2 | \([1, 1, 0, -1267, -14681]\) | \(169204136291/32906250\) | \(43798218750\) | \([2]\) | \(41472\) | \(0.75880\) |
Rank
sage: E.rank()
The elliptic curves in class 47190i have rank \(1\).
Complex multiplication
The elliptic curves in class 47190i do not have complex multiplication.Modular form 47190.2.a.i
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.