magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -300, 0, 12475, 0, -172920, 0, 1221495, 0, -5226012, 0, 14855725, 0, -29715200, 0, 43459480, 0, -47720380, 0, 40060019, 0, -26013000, 0, 13147875, 0, -5178240, 0, 1582240, 0, -371008, 0, 65450, 0, -8400, 0, 740, 0, -40, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 40*x^38 + 740*x^36 - 8400*x^34 + 65450*x^32 - 371008*x^30 + 1582240*x^28 - 5178240*x^26 + 13147875*x^24 - 26013000*x^22 + 40060019*x^20 - 47720380*x^18 + 43459480*x^16 - 29715200*x^14 + 14855725*x^12 - 5226012*x^10 + 1221495*x^8 - 172920*x^6 + 12475*x^4 - 300*x^2 + 1)
gp: K = bnfinit(x^40 - 40*x^38 + 740*x^36 - 8400*x^34 + 65450*x^32 - 371008*x^30 + 1582240*x^28 - 5178240*x^26 + 13147875*x^24 - 26013000*x^22 + 40060019*x^20 - 47720380*x^18 + 43459480*x^16 - 29715200*x^14 + 14855725*x^12 - 5226012*x^10 + 1221495*x^8 - 172920*x^6 + 12475*x^4 - 300*x^2 + 1, 1)
\( x^{40} - 40 x^{38} + 740 x^{36} - 8400 x^{34} + 65450 x^{32} - 371008 x^{30} + 1582240 x^{28} - 5178240 x^{26} + 13147875 x^{24} - 26013000 x^{22} + 40060019 x^{20} - 47720380 x^{18} + 43459480 x^{16} - 29715200 x^{14} + 14855725 x^{12} - 5226012 x^{10} + 1221495 x^{8} - 172920 x^{6} + 12475 x^{4} - 300 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
| Degree: | | $40$ |
|
| Signature: | | $[40, 0]$ |
|
| Discriminant: | | \(10240000000000000000000000000000000000000000000000000000000000000000000000=2^{80}\cdot 5^{70}\) | magma: Discriminant(Integers(K)); |
| Root discriminant: | | $66.87$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) |
| Ramified primes: | | $2, 5$ | magma: PrimeDivisors(Discriminant(Integers(K))); gp: factor(abs(K.disc))[,1]~ |
| This field is Galois and abelian over $\Q$. |
| Conductor: | | \(200=2^{3}\cdot 5^{2}\) |
| Dirichlet character group:
| |
$\lbrace$$\chi_{200}(1,·)$, $\chi_{200}(3,·)$, $\chi_{200}(7,·)$, $\chi_{200}(9,·)$, $\chi_{200}(141,·)$, $\chi_{200}(143,·)$, $\chi_{200}(147,·)$, $\chi_{200}(149,·)$, $\chi_{200}(23,·)$, $\chi_{200}(127,·)$, $\chi_{200}(27,·)$, $\chi_{200}(29,·)$, $\chi_{200}(161,·)$, $\chi_{200}(163,·)$, $\chi_{200}(167,·)$, $\chi_{200}(129,·)$, $\chi_{200}(41,·)$, $\chi_{200}(43,·)$, $\chi_{200}(47,·)$, $\chi_{200}(49,·)$, $\chi_{200}(181,·)$, $\chi_{200}(183,·)$, $\chi_{200}(187,·)$, $\chi_{200}(61,·)$, $\chi_{200}(63,·)$, $\chi_{200}(67,·)$, $\chi_{200}(69,·)$, $\chi_{200}(81,·)$, $\chi_{200}(83,·)$, $\chi_{200}(87,·)$, $\chi_{200}(89,·)$, $\chi_{200}(101,·)$, $\chi_{200}(103,·)$, $\chi_{200}(107,·)$, $\chi_{200}(109,·)$, $\chi_{200}(189,·)$, $\chi_{200}(169,·)$, $\chi_{200}(121,·)$, $\chi_{200}(123,·)$, $\chi_{200}(21,·)$$\rbrace$
|
| This is not a CM field. |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
Trivial group, which has order $1$
(assuming GRH)
sage: K.class_group().invariants()
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | | $39$
|
|
| Torsion generator: | | \( -1 \) (order $2$)
| magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() |
| Fundamental units: | | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
(assuming GRH)
| magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() |
| Regulator: | | \( 759341976624057400000000 \)
(assuming GRH)
|
|
$C_2\times C_{20}$ (as 40T2):
sage: K.galois_group(type='pari')
|
\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.8000.1, \(\Q(\zeta_{20})^+\), 5.5.390625.1, \(\Q(\zeta_{40})^+\), 10.10.5000000000000000.1, \(\Q(\zeta_{25})^+\), 10.10.25000000000000000.1, 20.20.625000000000000000000000000000000.1, 20.20.3125000000000000000000000000000000.1, \(\Q(\zeta_{100})^+\)
|
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| $p$ |
2 |
3 |
5 |
7 |
11 |
13 |
17 |
19 |
23 |
29 |
31 |
37 |
41 |
43 |
47 |
53 |
59 |
|---|
| Cycle type |
R |
$20^{2}$ |
R |
${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ |
${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ |
$20^{2}$ |
$20^{2}$ |
${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ |
$20^{2}$ |
${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ |
$20^{2}$ |
${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ |
${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ |
$20^{2}$ |
$20^{2}$ |
${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$ |
|---|
In the table, R denotes a ramified prime.
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
