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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 109554.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
109554.r1 | 109554q3 | \([1, 1, 1, -411328, 101366237]\) | \(8671983378625/82308\) | \(73048652975748\) | \([2]\) | \(1101600\) | \(1.8225\) | |
109554.r2 | 109554q4 | \([1, 1, 1, -401718, 106340373]\) | \(-8078253774625/846825858\) | \(-751561066140983298\) | \([2]\) | \(2203200\) | \(2.1691\) | |
109554.r3 | 109554q1 | \([1, 1, 1, -7708, -23107]\) | \(57066625/32832\) | \(29138520854592\) | \([2]\) | \(367200\) | \(1.2732\) | \(\Gamma_0(N)\)-optimal |
109554.r4 | 109554q2 | \([1, 1, 1, 30732, -146115]\) | \(3616805375/2105352\) | \(-1868507649800712\) | \([2]\) | \(734400\) | \(1.6198\) |
Rank
sage: E.rank()
The elliptic curves in class 109554.r have rank \(0\).
Complex multiplication
The elliptic curves in class 109554.r do not have complex multiplication.Modular form 109554.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.