Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 28050cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28050.cx8 | 28050cm1 | \([1, 1, 1, 1084312, -183437719]\) | \(9023321954633914439/6156756739584000\) | \(-96199324056000000000\) | \([2]\) | \(1327104\) | \(2.5231\) | \(\Gamma_0(N)\)-optimal |
28050.cx7 | 28050cm2 | \([1, 1, 1, -4747688, -1536461719]\) | \(757443433548897303481/373234243041000000\) | \(5831785047515625000000\) | \([2, 2]\) | \(2654208\) | \(2.8696\) | |
28050.cx6 | 28050cm3 | \([1, 1, 1, -19520063, -34012480219]\) | \(-52643812360427830814761/1504091705903677440\) | \(-23501432904744960000000\) | \([2]\) | \(3981312\) | \(3.0724\) | |
28050.cx5 | 28050cm4 | \([1, 1, 1, -40684688, 98799642281]\) | \(476646772170172569823801/5862293314453125000\) | \(91598333038330078125000\) | \([2]\) | \(5308416\) | \(3.2162\) | |
28050.cx4 | 28050cm5 | \([1, 1, 1, -62122688, -188349461719]\) | \(1696892787277117093383481/1440538624914939000\) | \(22508416014295921875000\) | \([2]\) | \(5308416\) | \(3.2162\) | |
28050.cx3 | 28050cm6 | \([1, 1, 1, -314432063, -2146172224219]\) | \(220031146443748723000125481/172266701724057600\) | \(2691667214438400000000\) | \([2, 2]\) | \(7962624\) | \(3.4189\) | |
28050.cx2 | 28050cm7 | \([1, 1, 1, -316544063, -2115881920219]\) | \(224494757451893010998773801/6152490825146276160000\) | \(96132669142910565000000000\) | \([2]\) | \(15925248\) | \(3.7655\) | |
28050.cx1 | 28050cm8 | \([1, 1, 1, -5030912063, -137348787904219]\) | \(901247067798311192691198986281/552431869440\) | \(8631747960000000\) | \([2]\) | \(15925248\) | \(3.7655\) |
Rank
sage: E.rank()
The elliptic curves in class 28050cm have rank \(0\).
Complex multiplication
The elliptic curves in class 28050cm do not have complex multiplication.Modular form 28050.2.a.cm
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.