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SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 162624hl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162624.v4 | 162624hl1 | \([0, -1, 0, 9156, -140490]\) | \(748613312/505197\) | \(-57279187361088\) | \([2]\) | \(491520\) | \(1.3286\) | \(\Gamma_0(N)\)-optimal |
162624.v3 | 162624hl2 | \([0, -1, 0, -39849, -1130391]\) | \(964430272/480249\) | \(3484837473030144\) | \([2, 2]\) | \(983040\) | \(1.6752\) | |
162624.v2 | 162624hl3 | \([0, -1, 0, -344769, 77234049]\) | \(78073482824/922383\) | \(53544804347510784\) | \([2]\) | \(1966080\) | \(2.0218\) | |
162624.v1 | 162624hl4 | \([0, -1, 0, -519009, -143632575]\) | \(266344154504/237699\) | \(13798548378058752\) | \([2]\) | \(1966080\) | \(2.0218\) |
Rank
sage: E.rank()
The elliptic curves in class 162624hl have rank \(0\).
Complex multiplication
The elliptic curves in class 162624hl do not have complex multiplication.Modular form 162624.2.a.hl
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.