Normalized defining polynomial
\( x^{32} + 8 x^{30} + 44 x^{28} + 208 x^{26} + 911 x^{24} + 2776 x^{22} + 7272 x^{20} + 17024 x^{18} + 34257 x^{16} + 47152 x^{14} + 59688 x^{12} + 65552 x^{10} + 50831 x^{8} + 3464 x^{6} + 236 x^{4} + 16 x^{2} + 1 \)
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: | \(203282392447840896882957090816000000000000000000000000=2^{96}\cdot 3^{16}\cdot 5^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $46.33$ | 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)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(240=2^{4}\cdot 3\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(131,·)$, $\chi_{240}(133,·)$, $\chi_{240}(11,·)$, $\chi_{240}(13,·)$, $\chi_{240}(143,·)$, $\chi_{240}(23,·)$, $\chi_{240}(157,·)$, $\chi_{240}(37,·)$, $\chi_{240}(167,·)$, $\chi_{240}(169,·)$, $\chi_{240}(47,·)$, $\chi_{240}(49,·)$, $\chi_{240}(179,·)$, $\chi_{240}(181,·)$, $\chi_{240}(59,·)$, $\chi_{240}(61,·)$, $\chi_{240}(191,·)$, $\chi_{240}(193,·)$, $\chi_{240}(71,·)$, $\chi_{240}(73,·)$, $\chi_{240}(203,·)$, $\chi_{240}(83,·)$, $\chi_{240}(217,·)$, $\chi_{240}(97,·)$, $\chi_{240}(227,·)$, $\chi_{240}(229,·)$, $\chi_{240}(107,·)$, $\chi_{240}(109,·)$, $\chi_{240}(239,·)$, $\chi_{240}(119,·)$, $\chi_{240}(121,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{1041753338401} a^{26} + \frac{79843284091}{1041753338401} a^{24} + \frac{486571795735}{1041753338401} a^{22} + \frac{212202007274}{1041753338401} a^{20} - \frac{505807280647}{1041753338401} a^{18} - \frac{402287136637}{1041753338401} a^{16} + \frac{88417645693}{1041753338401} a^{14} + \frac{432555802638}{1041753338401} a^{12} - \frac{71680044697}{1041753338401} a^{10} - \frac{115313573804}{1041753338401} a^{8} + \frac{285663179810}{1041753338401} a^{6} - \frac{421135329286}{1041753338401} a^{4} + \frac{96513802723}{1041753338401} a^{2} + \frac{337948059055}{1041753338401}$, $\frac{1}{1041753338401} a^{27} + \frac{79843284091}{1041753338401} a^{25} + \frac{486571795735}{1041753338401} a^{23} + \frac{212202007274}{1041753338401} a^{21} - \frac{505807280647}{1041753338401} a^{19} - \frac{402287136637}{1041753338401} a^{17} + \frac{88417645693}{1041753338401} a^{15} + \frac{432555802638}{1041753338401} a^{13} - \frac{71680044697}{1041753338401} a^{11} - \frac{115313573804}{1041753338401} a^{9} + \frac{285663179810}{1041753338401} a^{7} - \frac{421135329286}{1041753338401} a^{5} + \frac{96513802723}{1041753338401} a^{3} + \frac{337948059055}{1041753338401} a$, $\frac{1}{1041753338401} a^{28} + \frac{25054628700}{1041753338401} a^{24} + \frac{110659118192}{1041753338401} a^{22} + \frac{384180371535}{1041753338401} a^{20} + \frac{402287135589}{1041753338401} a^{18} - \frac{16052180812}{1041753338401} a^{16} - \frac{323269475077}{1041753338401} a^{14} + \frac{270132982402}{1041753338401} a^{12} - \frac{366695375033}{1041753338401} a^{10} + \frac{284680379083}{1041753338401} a^{8} - \frac{119244722731}{1041753338401} a^{6} - \frac{282094818063}{1041753338401} a^{4} + \frac{334503429196}{1041753338401} a^{2} - \frac{125192178357}{1041753338401}$, $\frac{1}{1041753338401} a^{29} + \frac{25054628700}{1041753338401} a^{25} + \frac{110659118192}{1041753338401} a^{23} + \frac{384180371535}{1041753338401} a^{21} + \frac{402287135589}{1041753338401} a^{19} - \frac{16052180812}{1041753338401} a^{17} - \frac{323269475077}{1041753338401} a^{15} + \frac{270132982402}{1041753338401} a^{13} - \frac{366695375033}{1041753338401} a^{11} + \frac{284680379083}{1041753338401} a^{9} - \frac{119244722731}{1041753338401} a^{7} - \frac{282094818063}{1041753338401} a^{5} + \frac{334503429196}{1041753338401} a^{3} - \frac{125192178357}{1041753338401} a$, $\frac{1}{1041753338401} a^{30} + \frac{182905806350}{1041753338401} a^{20} + \frac{234369048594}{1041753338401} a^{10} - \frac{415161683759}{1041753338401}$, $\frac{1}{1041753338401} a^{31} + \frac{182905806350}{1041753338401} a^{21} + \frac{234369048594}{1041753338401} a^{11} - \frac{415161683759}{1041753338401} a$
Class group and class number
$C_{8}\times C_{80}$, which has order $640$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{171204088}{1041753338401} a^{30} - \frac{941622484}{1041753338401} a^{28} - \frac{4451306288}{1041753338401} a^{26} - \frac{19495865521}{1041753338401} a^{24} - \frac{81838660680}{1041753338401} a^{22} - \frac{155624515992}{1041753338401} a^{20} - \frac{364322299264}{1041753338401} a^{18} - \frac{733117305327}{1041753338401} a^{16} - \frac{1009076894672}{1041753338401} a^{14} + \frac{919609660376}{1041753338401} a^{12} - \frac{1402846297072}{1041753338401} a^{10} - \frac{1087809374641}{1041753338401} a^{8} - \frac{74131370104}{1041753338401} a^{6} - \frac{5050520596}{1041753338401} a^{4} - \frac{15286898838064}{1041753338401} a^{2} - \frac{21400511}{1041753338401} \) (order $10$) | 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: | \( 164479581001.023 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_4^2$ (as 32T36):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2\times C_4^2$ |
| Character table for $C_2\times C_4^2$ 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 | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$ |
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 | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5 | Data not computed | ||||||