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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 73920ie
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
73920.hx3 | 73920ie1 | \([0, 1, 0, -63425, 4216383]\) | \(107639597521009/32699842560\) | \(8572067528048640\) | \([2]\) | \(491520\) | \(1.7623\) | \(\Gamma_0(N)\)-optimal |
73920.hx2 | 73920ie2 | \([0, 1, 0, -391105, -91007425]\) | \(25238585142450289/995844326400\) | \(261054615099801600\) | \([2, 2]\) | \(983040\) | \(2.1088\) | |
73920.hx4 | 73920ie3 | \([0, 1, 0, 172095, -331043265]\) | \(2150235484224911/181905111732960\) | \(-47685333610125066240\) | \([4]\) | \(1966080\) | \(2.4554\) | |
73920.hx1 | 73920ie4 | \([0, 1, 0, -6197185, -5940052417]\) | \(100407751863770656369/166028940000\) | \(43523490447360000\) | \([2]\) | \(1966080\) | \(2.4554\) |
Rank
sage: E.rank()
The elliptic curves in class 73920ie have rank \(1\).
Complex multiplication
The elliptic curves in class 73920ie do not have complex multiplication.Modular form 73920.2.a.ie
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.