Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 106575r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
106575.cg5 | 106575r1 | \([1, 1, 0, -960425, -388284000]\) | \(-53297461115137/4513839183\) | \(-8297635406886984375\) | \([2]\) | \(2359296\) | \(2.3752\) | \(\Gamma_0(N)\)-optimal |
106575.cg4 | 106575r2 | \([1, 1, 0, -15666550, -23873965625]\) | \(231331938231569617/1472026689\) | \(2705976061471265625\) | \([2, 2]\) | \(4718592\) | \(2.7218\) | |
106575.cg3 | 106575r3 | \([1, 1, 0, -15966675, -22912065000]\) | \(244883173420511137/18418027974129\) | \(33857227705129730015625\) | \([2, 2]\) | \(9437184\) | \(3.0683\) | |
106575.cg1 | 106575r4 | \([1, 1, 0, -250664425, -1527625367750]\) | \(947531277805646290177/38367\) | \(70528737234375\) | \([2]\) | \(9437184\) | \(3.0683\) | |
106575.cg6 | 106575r5 | \([1, 1, 0, 15289200, -101645614125]\) | \(215015459663151503/2552757445339983\) | \(-4692646260731307186984375\) | \([2]\) | \(18874368\) | \(3.4149\) | |
106575.cg2 | 106575r6 | \([1, 1, 0, -52024550, 117389126625]\) | \(8471112631466271697/1662662681263647\) | \(3056415652937293842234375\) | \([2]\) | \(18874368\) | \(3.4149\) |
Rank
sage: E.rank()
The elliptic curves in class 106575r have rank \(1\).
Complex multiplication
The elliptic curves in class 106575r do not have complex multiplication.Modular form 106575.2.a.r
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.