magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 210, 0, 7315, 0, 100947, 0, 735471, 0, 3268760, 0, 9657700, 0, 20058300, 0, 30421755, 0, 34597290, 0, 30045015, 0, 20160075, 0, 10518300, 0, 4272048, 0, 1344904, 0, 324632, 0, 58905, 0, 7770, 0, 703, 0, 39, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 39*x^38 + 703*x^36 + 7770*x^34 + 58905*x^32 + 324632*x^30 + 1344904*x^28 + 4272048*x^26 + 10518300*x^24 + 20160075*x^22 + 30045015*x^20 + 34597290*x^18 + 30421755*x^16 + 20058300*x^14 + 9657700*x^12 + 3268760*x^10 + 735471*x^8 + 100947*x^6 + 7315*x^4 + 210*x^2 + 1)
gp: K = bnfinit(x^40 + 39*x^38 + 703*x^36 + 7770*x^34 + 58905*x^32 + 324632*x^30 + 1344904*x^28 + 4272048*x^26 + 10518300*x^24 + 20160075*x^22 + 30045015*x^20 + 34597290*x^18 + 30421755*x^16 + 20058300*x^14 + 9657700*x^12 + 3268760*x^10 + 735471*x^8 + 100947*x^6 + 7315*x^4 + 210*x^2 + 1, 1)
\( x^{40} + 39 x^{38} + 703 x^{36} + 7770 x^{34} + 58905 x^{32} + 324632 x^{30} + 1344904 x^{28} + 4272048 x^{26} + 10518300 x^{24} + 20160075 x^{22} + 30045015 x^{20} + 34597290 x^{18} + 30421755 x^{16} + 20058300 x^{14} + 9657700 x^{12} + 3268760 x^{10} + 735471 x^{8} + 100947 x^{6} + 7315 x^{4} + 210 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
| Degree: | | $40$ |
|
| Signature: | | $[0, 20]$ |
|
| Discriminant: | | \(21231778848647571139402736714989215192335577851351941759145128045204996096=2^{40}\cdot 41^{38}\) | magma: Discriminant(Integers(K)); |
| Root discriminant: | | $68.10$ | 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: | | $2, 41$ | magma: PrimeDivisors(Discriminant(Integers(K))); gp: factor(abs(K.disc))[,1]~ |
| This field is Galois and abelian over $\Q$. |
| Conductor: | | \(164=2^{2}\cdot 41\) |
| Dirichlet character group:
| |
$\lbrace$$\chi_{164}(1,·)$, $\chi_{164}(131,·)$, $\chi_{164}(133,·)$, $\chi_{164}(9,·)$, $\chi_{164}(139,·)$, $\chi_{164}(141,·)$, $\chi_{164}(143,·)$, $\chi_{164}(21,·)$, $\chi_{164}(23,·)$, $\chi_{164}(25,·)$, $\chi_{164}(155,·)$, $\chi_{164}(5,·)$, $\chi_{164}(33,·)$, $\chi_{164}(163,·)$, $\chi_{164}(37,·)$, $\chi_{164}(39,·)$, $\chi_{164}(31,·)$, $\chi_{164}(43,·)$, $\chi_{164}(45,·)$, $\chi_{164}(49,·)$, $\chi_{164}(51,·)$, $\chi_{164}(57,·)$, $\chi_{164}(159,·)$, $\chi_{164}(61,·)$, $\chi_{164}(73,·)$, $\chi_{164}(77,·)$, $\chi_{164}(81,·)$, $\chi_{164}(83,·)$, $\chi_{164}(87,·)$, $\chi_{164}(91,·)$, $\chi_{164}(59,·)$, $\chi_{164}(103,·)$, $\chi_{164}(105,·)$, $\chi_{164}(107,·)$, $\chi_{164}(113,·)$, $\chi_{164}(115,·)$, $\chi_{164}(119,·)$, $\chi_{164}(121,·)$, $\chi_{164}(125,·)$, $\chi_{164}(127,·)$$\rbrace$
|
| This is a CM field. |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
Not computed
sage: K.class_group().invariants()
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | | $19$
|
|
| Torsion generator: | | \( a^{39} + 38 a^{37} + 666 a^{35} + 7140 a^{33} + 52360 a^{31} + 278256 a^{29} + 1107568 a^{27} + 3365856 a^{25} + 7888725 a^{23} + 14307150 a^{21} + 20030010 a^{19} + 21474180 a^{17} + 17383860 a^{15} + 10400600 a^{13} + 4457400 a^{11} + 1307504 a^{9} + 245157 a^{7} + 26334 a^{5} + 1330 a^{3} + 20 a \) (order $4$)
| magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() |
| Fundamental units: | | Not computed
| magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() |
| Regulator: | | Not computed
|
|
$C_2\times C_{20}$ (as 40T2):
sage: K.galois_group(type='pari')
|
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-41}) \), \(\Q(i, \sqrt{41})\), 4.0.1102736.1, 4.4.68921.1, 5.5.2825761.1, 8.0.1216026685696.1, 10.0.8176563434619904.1, 10.10.327381934393961.1, 10.0.335239100819416064.1, 20.0.112385254718210608313247485941252096.1, 20.0.4607795443446634940843146923591335936.1, \(\Q(\zeta_{41})^+\)
|
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 |
${\href{/LocalNumberField/3.4.0.1}{4} }^{10}$ |
${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ |
$20^{2}$ |
$20^{2}$ |
$20^{2}$ |
$20^{2}$ |
$20^{2}$ |
${\href{/LocalNumberField/23.10.0.1}{10} }^{4}$ |
$20^{2}$ |
${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$ |
R |
${\href{/LocalNumberField/43.10.0.1}{10} }^{4}$ |
$20^{2}$ |
$20^{2}$ |
${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$ |
|---|
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]])
