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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 270.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
270.b1 | 270d1 | \([1, -1, 0, -159, 813]\) | \(-16522921323/4000\) | \(-108000\) | \([3]\) | \(60\) | \(-0.045373\) | \(\Gamma_0(N)\)-optimal |
270.b2 | 270d2 | \([1, -1, 0, 66, 2708]\) | \(1601613/163840\) | \(-3224862720\) | \([]\) | \(180\) | \(0.50393\) |
Rank
sage: E.rank()
The elliptic curves in class 270.b have rank \(0\).
Complex multiplication
The elliptic curves in class 270.b do not have complex multiplication.Modular form 270.2.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 LMFDB 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 LMFDB labels.