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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 59150.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 59150.z1 | 59150o6 | \([1, 1, 0, -11536450, -15086719500]\) | \(2251439055699625/25088\) | \(1892109128000000\) | \([2]\) | \(1866240\) | \(2.5003\) | |
| 59150.z2 | 59150o5 | \([1, 1, 0, -720450, -236351500]\) | \(-548347731625/1835008\) | \(-138394267648000000\) | \([2]\) | \(933120\) | \(2.1537\) | |
| 59150.z3 | 59150o4 | \([1, 1, 0, -150075, -18404875]\) | \(4956477625/941192\) | \(70983656505125000\) | \([2]\) | \(622080\) | \(1.9510\) | |
| 59150.z4 | 59150o2 | \([1, 1, 0, -44450, 3586250]\) | \(128787625/98\) | \(7391051281250\) | \([2]\) | \(207360\) | \(1.4017\) | |
| 59150.z5 | 59150o1 | \([1, 1, 0, -2200, 79500]\) | \(-15625/28\) | \(-2111728937500\) | \([2]\) | \(103680\) | \(1.0551\) | \(\Gamma_0(N)\)-optimal |
| 59150.z6 | 59150o3 | \([1, 1, 0, 18925, -1673875]\) | \(9938375/21952\) | \(-1655595487000000\) | \([2]\) | \(311040\) | \(1.6044\) |
Rank
sage: E.rank()
The elliptic curves in class 59150.z have rank \(0\).
Complex multiplication
The elliptic curves in class 59150.z do not have complex multiplication.Modular form 59150.2.a.z
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.
