magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + x^38 - x^34 - x^32 + x^28 + x^26 - x^22 - x^20 - x^18 + x^14 + x^12 - x^8 - x^6 + x^2 + 1)
gp: K = bnfinit(x^40 + x^38 - x^34 - x^32 + x^28 + x^26 - x^22 - x^20 - x^18 + x^14 + x^12 - x^8 - x^6 + x^2 + 1, 1)
\( x^{40} + x^{38} - x^{34} - x^{32} + x^{28} + x^{26} - x^{22} - x^{20} - x^{18} + x^{14} + x^{12} - x^{8} - x^{6} + x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
| Degree: | | $40$ |
|
| Signature: | | $[0, 20]$ |
|
| Discriminant: | | \(118511797886229481159007653491590053243629014721874976833536=2^{40}\cdot 3^{20}\cdot 11^{36}\) | magma: Discriminant(Integers(K)); |
| Root discriminant: | | $29.98$ | 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: | | $2, 3, 11$ | magma: PrimeDivisors(Discriminant(Integers(K))); gp: factor(abs(K.disc))[,1]~ |
| This field is Galois and abelian over $\Q$. |
| Conductor: | | \(132=2^{2}\cdot 3\cdot 11\) |
| Dirichlet character group:
| |
$\lbrace$$\chi_{132}(1,·)$, $\chi_{132}(131,·)$, $\chi_{132}(5,·)$, $\chi_{132}(7,·)$, $\chi_{132}(13,·)$, $\chi_{132}(17,·)$, $\chi_{132}(19,·)$, $\chi_{132}(23,·)$, $\chi_{132}(25,·)$, $\chi_{132}(29,·)$, $\chi_{132}(31,·)$, $\chi_{132}(35,·)$, $\chi_{132}(37,·)$, $\chi_{132}(41,·)$, $\chi_{132}(43,·)$, $\chi_{132}(47,·)$, $\chi_{132}(49,·)$, $\chi_{132}(53,·)$, $\chi_{132}(59,·)$, $\chi_{132}(61,·)$, $\chi_{132}(65,·)$, $\chi_{132}(67,·)$, $\chi_{132}(71,·)$, $\chi_{132}(73,·)$, $\chi_{132}(79,·)$, $\chi_{132}(83,·)$, $\chi_{132}(85,·)$, $\chi_{132}(89,·)$, $\chi_{132}(91,·)$, $\chi_{132}(95,·)$, $\chi_{132}(97,·)$, $\chi_{132}(101,·)$, $\chi_{132}(103,·)$, $\chi_{132}(107,·)$, $\chi_{132}(109,·)$, $\chi_{132}(113,·)$, $\chi_{132}(115,·)$, $\chi_{132}(119,·)$, $\chi_{132}(125,·)$, $\chi_{132}(127,·)$$\rbrace$
|
| This is a CM field. |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
$C_{11}$, which has order $11$
(assuming GRH)
sage: K.class_group().invariants()
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | | $19$
|
|
| Torsion generator: | | \( a \) (order $132$)
| 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: | | \( 45016485591237.79 \)
(assuming GRH)
|
|
$C_2^2\times C_{10}$ (as 40T7):
sage: K.galois_group(type='pari')
|
\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-33}) \), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{33})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.0.303595776.1, 10.0.52089208083.1, 10.0.219503494144.1, 10.10.53339349076992.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{33})^+\), \(\Q(\zeta_{44})^+\), 10.0.586732839846912.1, 20.0.2845086159957207322343768064.1, \(\Q(\zeta_{33})\), 20.0.344255425354822086003595935744.2, \(\Q(\zeta_{44})\), 20.0.344255425354822086003595935744.1, 20.0.344255425354822086003595935744.3, \(\Q(\zeta_{132})^+\)
|
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 |
${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ |
R |
${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ |
${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$ |
${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ |
${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ |
${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$ |
|---|
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]])
