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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 43890s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43890.r4 | 43890s1 | \([1, 1, 0, 973, 70941]\) | \(101710228704839/2208184902000\) | \(-2208184902000\) | \([2]\) | \(110592\) | \(1.0489\) | \(\Gamma_0(N)\)-optimal |
43890.r3 | 43890s2 | \([1, 1, 0, -20807, 1085889]\) | \(996287603460985081/58993920562500\) | \(58993920562500\) | \([2, 2]\) | \(221184\) | \(1.3955\) | |
43890.r2 | 43890s3 | \([1, 1, 0, -62057, -4614861]\) | \(26430597275279725081/6198015706748250\) | \(6198015706748250\) | \([2]\) | \(442368\) | \(1.7420\) | |
43890.r1 | 43890s4 | \([1, 1, 0, -328037, 72178911]\) | \(3903860364706612157401/15001464843750\) | \(15001464843750\) | \([2]\) | \(442368\) | \(1.7420\) |
Rank
sage: E.rank()
The elliptic curves in class 43890s have rank \(2\).
Complex multiplication
The elliptic curves in class 43890s do not have complex multiplication.Modular form 43890.2.a.s
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.