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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 110b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
110.c1 | 110b1 | \([1, 0, 0, -1, 1]\) | \(-117649/440\) | \(-440\) | \([3]\) | \(4\) | \(-0.80711\) | \(\Gamma_0(N)\)-optimal |
110.c2 | 110b2 | \([1, 0, 0, 9, -25]\) | \(80062991/332750\) | \(-332750\) | \([]\) | \(12\) | \(-0.25780\) |
Rank
sage: E.rank()
The elliptic curves in class 110b have rank \(0\).
Complex multiplication
The elliptic curves in class 110b do not have complex multiplication.Modular form 110.2.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.