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SageMath
E = EllipticCurve("qa1")
E.isogeny_class()
Elliptic curves in class 100800.qa
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100800.qa1 | 100800oe4 | \([0, 0, 0, -1645500, -812446000]\) | \(2640279346000/3087\) | \(576108288000000\) | \([2]\) | \(1327104\) | \(2.1154\) | |
100800.qa2 | 100800oe3 | \([0, 0, 0, -102000, -12913000]\) | \(-10061824000/352947\) | \(-4116773808000000\) | \([2]\) | \(663552\) | \(1.7688\) | |
100800.qa3 | 100800oe2 | \([0, 0, 0, -25500, -502000]\) | \(9826000/5103\) | \(952342272000000\) | \([2]\) | \(442368\) | \(1.5661\) | |
100800.qa4 | 100800oe1 | \([0, 0, 0, 6000, -61000]\) | \(2048000/1323\) | \(-15431472000000\) | \([2]\) | \(221184\) | \(1.2195\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 100800.qa have rank \(0\).
Complex multiplication
The elliptic curves in class 100800.qa do not have complex multiplication.Modular form 100800.2.a.qa
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.