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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 28322r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28322.y1 | 28322r1 | \([1, -1, 1, -37225, -2602071]\) | \(9869198625/614656\) | \(355277029974272\) | \([2]\) | \(98304\) | \(1.5434\) | \(\Gamma_0(N)\)-optimal |
28322.y2 | 28322r2 | \([1, -1, 1, 29415, -10945399]\) | \(4869777375/92236816\) | \(-53313759310514192\) | \([2]\) | \(196608\) | \(1.8900\) |
Rank
sage: E.rank()
The elliptic curves in class 28322r have rank \(1\).
Complex multiplication
The elliptic curves in class 28322r do not have complex multiplication.Modular form 28322.2.a.r
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.