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SageMath
E = EllipticCurve("hw1")
E.isogeny_class()
Elliptic curves in class 62400.hw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
62400.hw1 | 62400hl4 | \([0, 1, 0, -30433, 1983263]\) | \(3044193988/85293\) | \(87340032000000\) | \([2]\) | \(262144\) | \(1.4541\) | |
62400.hw2 | 62400hl2 | \([0, 1, 0, -4433, -70737]\) | \(37642192/13689\) | \(3504384000000\) | \([2, 2]\) | \(131072\) | \(1.1076\) | |
62400.hw3 | 62400hl1 | \([0, 1, 0, -3933, -96237]\) | \(420616192/117\) | \(1872000000\) | \([2]\) | \(65536\) | \(0.76098\) | \(\Gamma_0(N)\)-optimal |
62400.hw4 | 62400hl3 | \([0, 1, 0, 13567, -484737]\) | \(269676572/257049\) | \(-263218176000000\) | \([2]\) | \(262144\) | \(1.4541\) |
Rank
sage: E.rank()
The elliptic curves in class 62400.hw have rank \(0\).
Complex multiplication
The elliptic curves in class 62400.hw do not have complex multiplication.Modular form 62400.2.a.hw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.