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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 11200.cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11200.cz1 | 11200co6 | \([0, -1, 0, -4368833, 3516221537]\) | \(2251439055699625/25088\) | \(102760448000000\) | \([2]\) | \(165888\) | \(2.2575\) | |
| 11200.cz2 | 11200co5 | \([0, -1, 0, -272833, 55101537]\) | \(-548347731625/1835008\) | \(-7516192768000000\) | \([2]\) | \(82944\) | \(1.9110\) | |
| 11200.cz3 | 11200co4 | \([0, -1, 0, -56833, 4293537]\) | \(4956477625/941192\) | \(3855122432000000\) | \([2]\) | \(55296\) | \(1.7082\) | |
| 11200.cz4 | 11200co2 | \([0, -1, 0, -16833, -834463]\) | \(128787625/98\) | \(401408000000\) | \([2]\) | \(18432\) | \(1.1589\) | |
| 11200.cz5 | 11200co1 | \([0, -1, 0, -833, -18463]\) | \(-15625/28\) | \(-114688000000\) | \([2]\) | \(9216\) | \(0.81236\) | \(\Gamma_0(N)\)-optimal |
| 11200.cz6 | 11200co3 | \([0, -1, 0, 7167, 389537]\) | \(9938375/21952\) | \(-89915392000000\) | \([2]\) | \(27648\) | \(1.3617\) |
Rank
sage: E.rank()
The elliptic curves in class 11200.cz have rank \(1\).
Complex multiplication
The elliptic curves in class 11200.cz do not have complex multiplication.Modular form 11200.2.a.cz
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
