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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 11109.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11109.d1 | 11109i5 | \([1, 0, 0, -414747, 102772518]\) | \(53297461115137/147\) | \(21761275683\) | \([2]\) | \(50688\) | \(1.6420\) | |
| 11109.d2 | 11109i4 | \([1, 0, 0, -25932, 1602855]\) | \(13027640977/21609\) | \(3198907525401\) | \([2, 2]\) | \(25344\) | \(1.2954\) | |
| 11109.d3 | 11109i3 | \([1, 0, 0, -20642, -1136307]\) | \(6570725617/45927\) | \(6798844274103\) | \([2]\) | \(25344\) | \(1.2954\) | |
| 11109.d4 | 11109i6 | \([1, 0, 0, -17997, 2604252]\) | \(-4354703137/17294403\) | \(-2560192322829267\) | \([2]\) | \(50688\) | \(1.6420\) | |
| 11109.d5 | 11109i2 | \([1, 0, 0, -2127, 7920]\) | \(7189057/3969\) | \(587554443441\) | \([2, 2]\) | \(12672\) | \(0.94880\) | |
| 11109.d6 | 11109i1 | \([1, 0, 0, 518, 1043]\) | \(103823/63\) | \(-9326261007\) | \([2]\) | \(6336\) | \(0.60223\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 11109.d have rank \(0\).
Complex multiplication
The elliptic curves in class 11109.d do not have complex multiplication.Modular form 11109.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
