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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 13200.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13200.cp1 | 13200ck3 | \([0, 1, 0, -32208, 2205588]\) | \(57736239625/255552\) | \(16355328000000\) | \([2]\) | \(41472\) | \(1.3878\) | |
| 13200.cp2 | 13200ck4 | \([0, 1, 0, -16208, 4413588]\) | \(-7357983625/127552392\) | \(-8163353088000000\) | \([2]\) | \(82944\) | \(1.7344\) | |
| 13200.cp3 | 13200ck1 | \([0, 1, 0, -2208, -38412]\) | \(18609625/1188\) | \(76032000000\) | \([2]\) | \(13824\) | \(0.83848\) | \(\Gamma_0(N)\)-optimal |
| 13200.cp4 | 13200ck2 | \([0, 1, 0, 1792, -158412]\) | \(9938375/176418\) | \(-11290752000000\) | \([2]\) | \(27648\) | \(1.1851\) |
Rank
sage: E.rank()
The elliptic curves in class 13200.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 13200.cp do not have complex multiplication.Modular form 13200.2.a.cp
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.
