Normalized defining polynomial
\( x^{32} + 27 x^{28} + 363 x^{24} + 9266 x^{20} + 125559 x^{16} + 108354 x^{12} + 65218 x^{8} + 24948 x^{4} + 6561 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(431022809469660770170681990696218941342911988389380096=2^{64}\cdot 3^{16}\cdot 13^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $47.43$ | 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: | $2, 3, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(312=2^{3}\cdot 3\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{312}(1,·)$, $\chi_{312}(259,·)$, $\chi_{312}(5,·)$, $\chi_{312}(265,·)$, $\chi_{312}(131,·)$, $\chi_{312}(151,·)$, $\chi_{312}(25,·)$, $\chi_{312}(155,·)$, $\chi_{312}(157,·)$, $\chi_{312}(287,·)$, $\chi_{312}(161,·)$, $\chi_{312}(47,·)$, $\chi_{312}(307,·)$, $\chi_{312}(53,·)$, $\chi_{312}(311,·)$, $\chi_{312}(31,·)$, $\chi_{312}(181,·)$, $\chi_{312}(73,·)$, $\chi_{312}(203,·)$, $\chi_{312}(77,·)$, $\chi_{312}(79,·)$, $\chi_{312}(209,·)$, $\chi_{312}(83,·)$, $\chi_{312}(187,·)$, $\chi_{312}(229,·)$, $\chi_{312}(103,·)$, $\chi_{312}(233,·)$, $\chi_{312}(235,·)$, $\chi_{312}(109,·)$, $\chi_{312}(239,·)$, $\chi_{312}(281,·)$, $\chi_{312}(125,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{4}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{5}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{6}$, $\frac{1}{27} a^{19} + \frac{1}{27} a^{15} + \frac{1}{9} a^{11} - \frac{2}{27} a^{7} + \frac{4}{27} a^{3}$, $\frac{1}{4941} a^{20} + \frac{88}{4941} a^{16} + \frac{121}{1647} a^{12} - \frac{1955}{4941} a^{8} - \frac{386}{4941} a^{4} - \frac{27}{61}$, $\frac{1}{4941} a^{21} + \frac{88}{4941} a^{17} + \frac{121}{1647} a^{13} - \frac{308}{4941} a^{9} - \frac{2033}{4941} a^{5} - \frac{20}{183} a$, $\frac{1}{4941} a^{22} + \frac{88}{4941} a^{18} - \frac{62}{1647} a^{14} - \frac{308}{4941} a^{10} - \frac{2033}{4941} a^{6} - \frac{121}{549} a^{2}$, $\frac{1}{4941} a^{23} + \frac{88}{4941} a^{19} - \frac{62}{1647} a^{15} - \frac{308}{4941} a^{11} - \frac{2033}{4941} a^{7} - \frac{121}{549} a^{3}$, $\frac{1}{1546533} a^{24} - \frac{76}{1546533} a^{20} + \frac{16675}{1546533} a^{16} - \frac{132308}{1546533} a^{12} - \frac{414328}{1546533} a^{8} - \frac{369299}{1546533} a^{4} - \frac{4356}{19093}$, $\frac{1}{1546533} a^{25} - \frac{76}{1546533} a^{21} + \frac{16675}{1546533} a^{17} - \frac{132308}{1546533} a^{13} + \frac{101183}{1546533} a^{9} + \frac{661723}{1546533} a^{5} + \frac{6025}{57279} a$, $\frac{1}{1546533} a^{26} - \frac{76}{1546533} a^{22} + \frac{16675}{1546533} a^{18} + \frac{39529}{1546533} a^{14} + \frac{101183}{1546533} a^{10} + \frac{661723}{1546533} a^{6} + \frac{37168}{171837} a^{2}$, $\frac{1}{1546533} a^{27} - \frac{76}{1546533} a^{23} + \frac{16675}{1546533} a^{19} + \frac{39529}{1546533} a^{15} + \frac{101183}{1546533} a^{11} + \frac{661723}{1546533} a^{7} + \frac{37168}{171837} a^{3}$, $\frac{1}{681679269207} a^{28} + \frac{158603}{681679269207} a^{24} + \frac{22259680}{681679269207} a^{20} + \frac{36253861327}{681679269207} a^{16} + \frac{16352504330}{681679269207} a^{12} - \frac{311840999840}{681679269207} a^{8} - \frac{2737898736}{8415793447} a^{4} - \frac{624101028}{8415793447}$, $\frac{1}{2045037807621} a^{29} + \frac{66598}{227226423069} a^{25} - \frac{1248836}{227226423069} a^{21} + \frac{714817232}{33525209961} a^{17} - \frac{4662898178}{227226423069} a^{13} + \frac{20801243095}{227226423069} a^{9} - \frac{384549041537}{2045037807621} a^{5} - \frac{848044784}{8415793447} a$, $\frac{1}{6135113422863} a^{30} + \frac{66598}{681679269207} a^{26} + \frac{134217319}{2045037807621} a^{22} - \frac{147200121589}{6135113422863} a^{18} + \frac{12030724889}{681679269207} a^{14} + \frac{247137293638}{2045037807621} a^{10} - \frac{2134896114686}{6135113422863} a^{6} + \frac{3112382555}{75742141023} a^{2}$, $\frac{1}{18405340268589} a^{31} - \frac{124727}{681679269207} a^{27} - \frac{179176550}{6135113422863} a^{23} - \frac{322617380998}{18405340268589} a^{19} - \frac{55473697403}{2045037807621} a^{15} + \frac{922497114322}{6135113422863} a^{11} - \frac{4280672199641}{18405340268589} a^{7} - \frac{55487422379}{227226423069} a^{3}$
Class group and class number
$C_{4}\times C_{4}\times C_{12}$, which has order $192$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{81714466}{2045037807621} a^{29} - \frac{27015025}{25247380341} a^{25} - \frac{3236399995}{227226423069} a^{21} - \frac{749370514412}{2045037807621} a^{17} - \frac{373207167370}{75742141023} a^{13} - \frac{685052406955}{227226423069} a^{9} + \frac{2203513790273}{2045037807621} a^{5} - \frac{2625860430}{8415793447} a \) (order $24$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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()
gp: K.fu
| |
| Regulator: | \( 4647831490592.4375 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3\times C_4$ (as 32T34):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^3\times C_4$ |
| Character table for $C_2^3\times C_4$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |