Normalized defining polynomial
\( x^{32} - 84 x^{28} + 5930 x^{24} - 162606 x^{20} + 2174199 x^{16} - 17206446 x^{12} + 89094515 x^{8} - 280197189 x^{4} + 519885601 \)
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: | \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.29$ | 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: | $2, 3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(391,·)$, $\chi_{840}(139,·)$, $\chi_{840}(533,·)$, $\chi_{840}(407,·)$, $\chi_{840}(797,·)$, $\chi_{840}(799,·)$, $\chi_{840}(293,·)$, $\chi_{840}(167,·)$, $\chi_{840}(169,·)$, $\chi_{840}(811,·)$, $\chi_{840}(559,·)$, $\chi_{840}(181,·)$, $\chi_{840}(631,·)$, $\chi_{840}(827,·)$, $\chi_{840}(769,·)$, $\chi_{840}(323,·)$, $\chi_{840}(197,·)$, $\chi_{840}(713,·)$, $\chi_{840}(587,·)$, $\chi_{840}(589,·)$, $\chi_{840}(83,·)$, $\chi_{840}(601,·)$, $\chi_{840}(349,·)$, $\chi_{840}(421,·)$, $\chi_{840}(743,·)$, $\chi_{840}(617,·)$, $\chi_{840}(113,·)$, $\chi_{840}(211,·)$, $\chi_{840}(503,·)$, $\chi_{840}(377,·)$, $\chi_{840}(379,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{20} + \frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{21} + \frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{22} + \frac{1}{3} a^{14} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{23} + \frac{1}{3} a^{15} + \frac{1}{3} a^{7}$, $\frac{1}{957} a^{24} + \frac{35}{957} a^{20} + \frac{65}{957} a^{16} + \frac{332}{957} a^{12} - \frac{67}{957} a^{8} - \frac{460}{957} a^{4} - \frac{266}{957}$, $\frac{1}{957} a^{25} + \frac{35}{957} a^{21} + \frac{65}{957} a^{17} + \frac{332}{957} a^{13} - \frac{67}{957} a^{9} - \frac{460}{957} a^{5} - \frac{266}{957} a$, $\frac{1}{957} a^{26} + \frac{35}{957} a^{22} + \frac{65}{957} a^{18} + \frac{332}{957} a^{14} - \frac{67}{957} a^{10} - \frac{460}{957} a^{6} - \frac{266}{957} a^{2}$, $\frac{1}{957} a^{27} + \frac{35}{957} a^{23} + \frac{65}{957} a^{19} + \frac{332}{957} a^{15} - \frac{67}{957} a^{11} - \frac{460}{957} a^{7} - \frac{266}{957} a^{3}$, $\frac{1}{237854587632519778405860903} a^{28} - \frac{74869830573914549567609}{237854587632519778405860903} a^{24} + \frac{12867451907543395593305554}{79284862544173259468620301} a^{20} - \frac{192189275493533337898907}{4172887502324908393085279} a^{16} + \frac{137072208671800435150280}{398416394694337987279499} a^{12} + \frac{38209518068678058504209507}{79284862544173259468620301} a^{8} + \frac{103304093955127036247390855}{237854587632519778405860903} a^{4} + \frac{114487810669644294677950028}{237854587632519778405860903}$, $\frac{1}{35916042732510486539284996353} a^{29} - \frac{2924611980145394799073958}{11972014244170162179761665451} a^{25} - \frac{153908089447060720756649821}{3265094793864589685389545123} a^{21} + \frac{119887761355055705290135048}{1890318038553183502067631387} a^{17} - \frac{847330419994120427598472}{16407511526957737112510277} a^{13} - \frac{14525234324914459562669633516}{35916042732510486539284996353} a^{9} + \frac{3350397731768255208771625110}{11972014244170162179761665451} a^{5} - \frac{1139406018594788601933929027}{35916042732510486539284996353} a$, $\frac{1}{5423322452609083467432034449303} a^{30} - \frac{77984262413814123198833616}{1807774150869694489144011483101} a^{26} + \frac{421418260510672954061122044115}{5423322452609083467432034449303} a^{22} - \frac{46648200220696904355791691859}{285438023821530708812212339437} a^{18} - \frac{13199262573515305162770896561}{27252876646276801343879570097} a^{14} + \frac{419582253958208631137340348527}{5423322452609083467432034449303} a^{10} - \frac{73830470400438694953674289021}{164343104624517680831273771191} a^{6} - \frac{2503252853224941849807922128457}{5423322452609083467432034449303} a^{2}$, $\frac{1}{818921690343971603582237201844753} a^{31} - \frac{368589187290470726991417022483}{818921690343971603582237201844753} a^{27} + \frac{4279274388357790280104521516768}{272973896781323867860745733948251} a^{23} + \frac{230418440725621747172734475325}{14367047199017045676881354418329} a^{19} - \frac{421214039645025598763679439557}{1371728124529265667641938361549} a^{15} + \frac{124064006644218637538816506684488}{272973896781323867860745733948251} a^{11} + \frac{132659293481746798698829970235488}{818921690343971603582237201844753} a^{7} - \frac{278730009705392711393214040375454}{818921690343971603582237201844753} a^{3}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{40}$, which has order $1280$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{2960644082335055}{37159280789898107522671} a^{29} - \frac{594716387905978306}{111477842369694322568013} a^{25} + \frac{42061666494004063375}{111477842369694322568013} a^{21} - \frac{36706906042738985675}{5867254861562859082527} a^{17} + \frac{5309321311837552238995}{111477842369694322568013} a^{13} - \frac{33853676452044864427610}{111477842369694322568013} a^{9} + \frac{10591923570274339227176}{10134349306335847506183} a^{5} - \frac{360363771576335250169900}{111477842369694322568013} a \) (order $8$) | 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: | \( 17282058402401.408 \) (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 | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$ |
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 | ||||||
| $3$ | 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |