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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 123210.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123210.q1 | 123210bd2 | \([1, -1, 0, -1755, 28701]\) | \(599188249/1000\) | \(998001000\) | \([]\) | \(77760\) | \(0.62238\) | |
123210.q2 | 123210bd1 | \([1, -1, 0, -90, -270]\) | \(81289/10\) | \(9980010\) | \([]\) | \(25920\) | \(0.073078\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 123210.q have rank \(0\).
Complex multiplication
The elliptic curves in class 123210.q do not have complex multiplication.Modular form 123210.2.a.q
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.