magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -120, 0, 4018, 0, -59241, 0, 470613, 0, -2269345, 0, 7199257, 0, -15882694, 0, 25334997, 0, -30037766, 0, 26988270, 0, -18613969, 0, 9927425, 0, -4102514, 0, 1308973, 0, -319177, 0, 58344, 0, -7735, 0, 702, 0, -39, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 39*x^38 + 702*x^36 - 7735*x^34 + 58344*x^32 - 319177*x^30 + 1308973*x^28 - 4102514*x^26 + 9927425*x^24 - 18613969*x^22 + 26988270*x^20 - 30037766*x^18 + 25334997*x^16 - 15882694*x^14 + 7199257*x^12 - 2269345*x^10 + 470613*x^8 - 59241*x^6 + 4018*x^4 - 120*x^2 + 1)
gp: K = bnfinit(x^40 - 39*x^38 + 702*x^36 - 7735*x^34 + 58344*x^32 - 319177*x^30 + 1308973*x^28 - 4102514*x^26 + 9927425*x^24 - 18613969*x^22 + 26988270*x^20 - 30037766*x^18 + 25334997*x^16 - 15882694*x^14 + 7199257*x^12 - 2269345*x^10 + 470613*x^8 - 59241*x^6 + 4018*x^4 - 120*x^2 + 1, 1)
\( x^{40} - 39 x^{38} + 702 x^{36} - 7735 x^{34} + 58344 x^{32} - 319177 x^{30} + 1308973 x^{28} - 4102514 x^{26} + 9927425 x^{24} - 18613969 x^{22} + 26988270 x^{20} - 30037766 x^{18} + 25334997 x^{16} - 15882694 x^{14} + 7199257 x^{12} - 2269345 x^{10} + 470613 x^{8} - 59241 x^{6} + 4018 x^{4} - 120 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
| Degree: | | $40$ |
|
| Signature: | | $[40, 0]$ |
|
| Discriminant: | | \(31654584865659568778929513407372752241664000000000000000000000000000000=2^{40}\cdot 5^{30}\cdot 11^{36}\) | magma: Discriminant(Integers(K)); |
| Root discriminant: | | $57.88$ | 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: | | $2, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K))); gp: factor(abs(K.disc))[,1]~ |
| This field is Galois and abelian over $\Q$. |
| Conductor: | | \(220=2^{2}\cdot 5\cdot 11\) |
| Dirichlet character group:
| |
$\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(3,·)$, $\chi_{220}(9,·)$, $\chi_{220}(139,·)$, $\chi_{220}(13,·)$, $\chi_{220}(17,·)$, $\chi_{220}(131,·)$, $\chi_{220}(23,·)$, $\chi_{220}(153,·)$, $\chi_{220}(27,·)$, $\chi_{220}(69,·)$, $\chi_{220}(163,·)$, $\chi_{220}(39,·)$, $\chi_{220}(169,·)$, $\chi_{220}(171,·)$, $\chi_{220}(173,·)$, $\chi_{220}(47,·)$, $\chi_{220}(49,·)$, $\chi_{220}(51,·)$, $\chi_{220}(181,·)$, $\chi_{220}(73,·)$, $\chi_{220}(57,·)$, $\chi_{220}(151,·)$, $\chi_{220}(19,·)$, $\chi_{220}(193,·)$, $\chi_{220}(67,·)$, $\chi_{220}(197,·)$, $\chi_{220}(201,·)$, $\chi_{220}(79,·)$, $\chi_{220}(203,·)$, $\chi_{220}(207,·)$, $\chi_{220}(141,·)$, $\chi_{220}(81,·)$, $\chi_{220}(211,·)$, $\chi_{220}(89,·)$, $\chi_{220}(219,·)$, $\chi_{220}(103,·)$, $\chi_{220}(147,·)$, $\chi_{220}(117,·)$, $\chi_{220}(217,·)$$\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: | | \( 33894240488185236000000 \)
(assuming GRH)
|
|
$C_2\times C_{20}$ (as 40T2):
sage: K.galois_group(type='pari')
|
\(\Q(\sqrt{11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{5}, \sqrt{11})\), 4.4.15125.1, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{11})^+\), 8.8.58564000000.1, \(\Q(\zeta_{44})^+\), 10.10.669871503125.1, 10.10.7545432611200000.1, 20.20.56933553290160450365440000000000.1, \(\Q(\zeta_{55})^+\), 20.20.1470391355634309152000000000000000.1
|
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 |
$20^{2}$ |
R |
$20^{2}$ |
$20^{2}$ |
${\href{/LocalNumberField/19.5.0.1}{5} }^{8}$ |
${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ |
${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ |
$20^{2}$ |
${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ |
$20^{2}$ |
$20^{2}$ |
${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$ |
|---|
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]])
