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SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 25200dx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25200.cr6 | 25200dx1 | \([0, 0, 0, 3525, -17750]\) | \(103823/63\) | \(-2939328000000\) | \([2]\) | \(32768\) | \(1.0817\) | \(\Gamma_0(N)\)-optimal |
25200.cr5 | 25200dx2 | \([0, 0, 0, -14475, -143750]\) | \(7189057/3969\) | \(185177664000000\) | \([2, 2]\) | \(65536\) | \(1.4282\) | |
25200.cr3 | 25200dx3 | \([0, 0, 0, -140475, 20142250]\) | \(6570725617/45927\) | \(2142770112000000\) | \([2]\) | \(131072\) | \(1.7748\) | |
25200.cr2 | 25200dx4 | \([0, 0, 0, -176475, -28493750]\) | \(13027640977/21609\) | \(1008189504000000\) | \([2, 2]\) | \(131072\) | \(1.7748\) | |
25200.cr4 | 25200dx5 | \([0, 0, 0, -122475, -46259750]\) | \(-4354703137/17294403\) | \(-806887666368000000\) | \([2]\) | \(262144\) | \(2.1214\) | |
25200.cr1 | 25200dx6 | \([0, 0, 0, -2822475, -1825127750]\) | \(53297461115137/147\) | \(6858432000000\) | \([2]\) | \(262144\) | \(2.1214\) |
Rank
sage: E.rank()
The elliptic curves in class 25200dx have rank \(1\).
Complex multiplication
The elliptic curves in class 25200dx do not have complex multiplication.Modular form 25200.2.a.dx
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.