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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 21168.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21168.r1 | 21168dr3 | \([0, 0, 0, -96579, -15317694]\) | \(-1167051/512\) | \(-43707095901536256\) | \([]\) | \(163296\) | \(1.9002\) | |
21168.r2 | 21168dr1 | \([0, 0, 0, -2499, 48706]\) | \(-132651/2\) | \(-26022076416\) | \([]\) | \(18144\) | \(0.80159\) | \(\Gamma_0(N)\)-optimal |
21168.r3 | 21168dr2 | \([0, 0, 0, 9261, 240786]\) | \(9261/8\) | \(-75880374829056\) | \([]\) | \(54432\) | \(1.3509\) |
Rank
sage: E.rank()
The elliptic curves in class 21168.r have rank \(0\).
Complex multiplication
The elliptic curves in class 21168.r do not have complex multiplication.Modular form 21168.2.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 LMFDB numbering.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.