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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 55440.dp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55440.dp1 | 55440ch4 | \([0, 0, 0, -653427, 190126386]\) | \(382704614800227/27778076480\) | \(2239512081841520640\) | \([2]\) | \(995328\) | \(2.2675\) | |
55440.dp2 | 55440ch2 | \([0, 0, 0, -124227, -16798654]\) | \(1917114236485083/7117764500\) | \(787167811584000\) | \([2]\) | \(331776\) | \(1.7182\) | |
55440.dp3 | 55440ch1 | \([0, 0, 0, -4227, -502654]\) | \(-75526045083/943250000\) | \(-104315904000000\) | \([2]\) | \(165888\) | \(1.3716\) | \(\Gamma_0(N)\)-optimal |
55440.dp4 | 55440ch3 | \([0, 0, 0, 37773, 13040946]\) | \(73929353373/954060800\) | \(-76917877663334400\) | \([2]\) | \(497664\) | \(1.9209\) |
Rank
sage: E.rank()
The elliptic curves in class 55440.dp have rank \(0\).
Complex multiplication
The elliptic curves in class 55440.dp do not have complex multiplication.Modular form 55440.2.a.dp
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.