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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 72128f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72128.bv2 | 72128f1 | \([0, -1, 0, 108127, -64410079]\) | \(4533086375/60669952\) | \(-1871120743228506112\) | \([2]\) | \(1032192\) | \(2.1869\) | \(\Gamma_0(N)\)-optimal |
72128.bv1 | 72128f2 | \([0, -1, 0, -1898913, -942289375]\) | \(24553362849625/1755162752\) | \(54130938376368422912\) | \([2]\) | \(2064384\) | \(2.5334\) |
Rank
sage: E.rank()
The elliptic curves in class 72128f have rank \(0\).
Complex multiplication
The elliptic curves in class 72128f do not have complex multiplication.Modular form 72128.2.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.