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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 117117bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117117.n2 | 117117bl1 | \([1, -1, 1, -13136, 618882]\) | \(-156503678869/11647251\) | \(-18654388615863\) | \([2]\) | \(248832\) | \(1.2952\) | \(\Gamma_0(N)\)-optimal |
117117.n1 | 117117bl2 | \([1, -1, 1, -213791, 38101236]\) | \(674733819141829/3361743\) | \(5384211291459\) | \([2]\) | \(497664\) | \(1.6418\) |
Rank
sage: E.rank()
The elliptic curves in class 117117bl have rank \(2\).
Complex multiplication
The elliptic curves in class 117117bl do not have complex multiplication.Modular form 117117.2.a.bl
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.