magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 600, 0, 17450, 0, 206465, 0, 1345115, 0, 5507149, 0, 15277900, 0, 30152050, 0, 43779420, 0, 47888645, 0, 40123776, 0, 26030250, 0, 13151125, 0, 5178645, 0, 1582270, 0, 371009, 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 + 371009*x^30 + 1582270*x^28 + 5178645*x^26 + 13151125*x^24 + 26030250*x^22 + 40123776*x^20 + 47888645*x^18 + 43779420*x^16 + 30152050*x^14 + 15277900*x^12 + 5507149*x^10 + 1345115*x^8 + 206465*x^6 + 17450*x^4 + 600*x^2 + 1)
gp: K = bnfinit(x^40 + 40*x^38 + 740*x^36 + 8400*x^34 + 65450*x^32 + 371009*x^30 + 1582270*x^28 + 5178645*x^26 + 13151125*x^24 + 26030250*x^22 + 40123776*x^20 + 47888645*x^18 + 43779420*x^16 + 30152050*x^14 + 15277900*x^12 + 5507149*x^10 + 1345115*x^8 + 206465*x^6 + 17450*x^4 + 600*x^2 + 1, 1)
\( x^{40} + 40 x^{38} + 740 x^{36} + 8400 x^{34} + 65450 x^{32} + 371009 x^{30} + 1582270 x^{28} + 5178645 x^{26} + 13151125 x^{24} + 26030250 x^{22} + 40123776 x^{20} + 47888645 x^{18} + 43779420 x^{16} + 30152050 x^{14} + 15277900 x^{12} + 5507149 x^{10} + 1345115 x^{8} + 206465 x^{6} + 17450 x^{4} + 600 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
| Degree: | | $40$ |
|
| Signature: | | $[0, 20]$ |
|
| Discriminant: | | \(32473210254684090614318847656250000000000000000000000000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{70}\) | magma: Discriminant(Integers(K)); |
| Root discriminant: | | $57.91$ | 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: | | $2, 3, 5$ | magma: PrimeDivisors(Discriminant(Integers(K))); gp: factor(abs(K.disc))[,1]~ |
| This field is Galois and abelian over $\Q$. |
| Conductor: | | \(300=2^{2}\cdot 3\cdot 5^{2}\) |
| Dirichlet character group:
| |
$\lbrace$$\chi_{300}(1,·)$, $\chi_{300}(259,·)$, $\chi_{300}(263,·)$, $\chi_{300}(137,·)$, $\chi_{300}(139,·)$, $\chi_{300}(143,·)$, $\chi_{300}(17,·)$, $\chi_{300}(19,·)$, $\chi_{300}(23,·)$, $\chi_{300}(287,·)$, $\chi_{300}(289,·)$, $\chi_{300}(91,·)$, $\chi_{300}(293,·)$, $\chi_{300}(167,·)$, $\chi_{300}(169,·)$, $\chi_{300}(257,·)$, $\chi_{300}(173,·)$, $\chi_{300}(47,·)$, $\chi_{300}(49,·)$, $\chi_{300}(53,·)$, $\chi_{300}(31,·)$, $\chi_{300}(151,·)$, $\chi_{300}(61,·)$, $\chi_{300}(181,·)$, $\chi_{300}(197,·)$, $\chi_{300}(199,·)$, $\chi_{300}(203,·)$, $\chi_{300}(77,·)$, $\chi_{300}(79,·)$, $\chi_{300}(83,·)$, $\chi_{300}(271,·)$, $\chi_{300}(227,·)$, $\chi_{300}(229,·)$, $\chi_{300}(233,·)$, $\chi_{300}(107,·)$, $\chi_{300}(109,·)$, $\chi_{300}(113,·)$, $\chi_{300}(211,·)$, $\chi_{300}(241,·)$, $\chi_{300}(121,·)$$\rbrace$
|
| This is a CM field. |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
$C_{237710}$, which has order $237710$
(assuming GRH)
sage: K.class_group().invariants()
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | | $19$
|
|
| Torsion generator: | | \( a^{25} + 25 a^{23} + 275 a^{21} + 1750 a^{19} + 7125 a^{17} + 19380 a^{15} + 35700 a^{13} + 44200 a^{11} + 35750 a^{9} + 17875 a^{7} + 5005 a^{5} + 650 a^{3} + 25 a \) (order $4$)
| 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: | | \( 64293715230028.38 \)
(assuming GRH)
|
|
$C_2\times C_{20}$ (as 40T2):
sage: K.galois_group(type='pari')
|
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 4.0.18000.1, \(\Q(\zeta_{15})^+\), 5.5.390625.1, 8.0.324000000.1, 10.0.156250000000000.1, \(\Q(\zeta_{25})^+\), 10.0.781250000000000.1, 20.0.610351562500000000000000000000.1, 20.0.180203247070312500000000000000000000.1, \(\Q(\zeta_{75})^+\)
|
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 |
R |
R |
${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ |
${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ |
$20^{2}$ |
$20^{2}$ |
${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ |
$20^{2}$ |
${\href{/LocalNumberField/29.5.0.1}{5} }^{8}$ |
${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ |
$20^{2}$ |
${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ |
$20^{2}$ |
$20^{2}$ |
${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$ |
|---|
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]])
