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SageMath
E = EllipticCurve("ci1")
E.isogeny_class()
Elliptic curves in class 12600.ci
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12600.ci1 | 12600cc3 | \([0, 0, 0, -67275, -6716250]\) | \(1443468546/7\) | \(163296000000\) | \([2]\) | \(32768\) | \(1.3520\) | |
12600.ci2 | 12600cc4 | \([0, 0, 0, -13275, 465750]\) | \(11090466/2401\) | \(56010528000000\) | \([2]\) | \(32768\) | \(1.3520\) | |
12600.ci3 | 12600cc2 | \([0, 0, 0, -4275, -101250]\) | \(740772/49\) | \(571536000000\) | \([2, 2]\) | \(16384\) | \(1.0054\) | |
12600.ci4 | 12600cc1 | \([0, 0, 0, 225, -6750]\) | \(432/7\) | \(-20412000000\) | \([2]\) | \(8192\) | \(0.65884\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12600.ci have rank \(0\).
Complex multiplication
The elliptic curves in class 12600.ci do not have complex multiplication.Modular form 12600.2.a.ci
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.