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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 169338b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169338.w2 | 169338b1 | \([1, 0, 0, -764, -33840]\) | \(-10218313/96192\) | \(-464300411328\) | \([2]\) | \(338688\) | \(0.92105\) | \(\Gamma_0(N)\)-optimal |
169338.w1 | 169338b2 | \([1, 0, 0, -21044, -1173576]\) | \(213525509833/669336\) | \(3230757028824\) | \([2]\) | \(677376\) | \(1.2676\) |
Rank
sage: E.rank()
The elliptic curves in class 169338b have rank \(1\).
Complex multiplication
The elliptic curves in class 169338b do not have complex multiplication.Modular form 169338.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.