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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 189.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
189.b1 | 189b3 | \([0, 0, 1, -3834, -91375]\) | \(35184082944/7\) | \(1240029\) | \([]\) | \(108\) | \(0.55863\) | |
189.b2 | 189b2 | \([0, 0, 1, -54, -88]\) | \(884736/343\) | \(6751269\) | \([3]\) | \(36\) | \(0.0093281\) | |
189.b3 | 189b1 | \([0, 0, 1, -24, 45]\) | \(56623104/7\) | \(189\) | \([3]\) | \(12\) | \(-0.53998\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 189.b have rank \(1\).
Complex multiplication
The elliptic curves in class 189.b do not have complex multiplication.Modular form 189.2.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.