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SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 55440dc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55440.b4 | 55440dc1 | \([0, 0, 0, -4203, -1109862]\) | \(-2749884201/176619520\) | \(-527383060807680\) | \([2]\) | \(196608\) | \(1.5047\) | \(\Gamma_0(N)\)-optimal |
55440.b3 | 55440dc2 | \([0, 0, 0, -188523, -31301478]\) | \(248158561089321/1859334400\) | \(5551942769049600\) | \([2, 2]\) | \(393216\) | \(1.8513\) | |
55440.b2 | 55440dc3 | \([0, 0, 0, -315243, 16066458]\) | \(1160306142246441/634128110000\) | \(1893496390410240000\) | \([2]\) | \(786432\) | \(2.1978\) | |
55440.b1 | 55440dc4 | \([0, 0, 0, -3010923, -2010932838]\) | \(1010962818911303721/57392720\) | \(171373743636480\) | \([2]\) | \(786432\) | \(2.1978\) |
Rank
sage: E.rank()
The elliptic curves in class 55440dc have rank \(1\).
Complex multiplication
The elliptic curves in class 55440dc do not have complex multiplication.Modular form 55440.2.a.dc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.