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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 75150b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75150.t2 | 75150b1 | \([1, -1, 0, -66567, -7422659]\) | \(-77325109990227/11923105280\) | \(-5030060040000000\) | \([]\) | \(414720\) | \(1.7418\) | \(\Gamma_0(N)\)-optimal |
75150.t1 | 75150b2 | \([1, -1, 0, -5577567, -5068691659]\) | \(-62394574179743883/167000\) | \(-51360328125000\) | \([]\) | \(1244160\) | \(2.2911\) |
Rank
sage: E.rank()
The elliptic curves in class 75150b have rank \(1\).
Complex multiplication
The elliptic curves in class 75150b do not have complex multiplication.Modular form 75150.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.