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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 171.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
171.b1 | 171b3 | \([0, 0, 1, -6924, 221760]\) | \(-50357871050752/19\) | \(-13851\) | \([3]\) | \(72\) | \(0.58275\) | |
171.b2 | 171b2 | \([0, 0, 1, -84, 315]\) | \(-89915392/6859\) | \(-5000211\) | \([3]\) | \(24\) | \(0.033439\) | |
171.b3 | 171b1 | \([0, 0, 1, 6, 0]\) | \(32768/19\) | \(-13851\) | \([]\) | \(8\) | \(-0.51587\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 171.b have rank \(1\).
Complex multiplication
The elliptic curves in class 171.b do not have complex multiplication.Modular form 171.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.