Normalized defining polynomial
\( x^{32} + 236 x^{28} + 10506 x^{24} + 137746 x^{20} + 691751 x^{16} + 1498330 x^{12} + 1339275 x^{8} + 393875 x^{4} + 625 \)
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$$\chi_{840}(1,·)$, $\chi_{840}(43,·)$, $\chi_{840}(139,·)$, $\chi_{840}(659,·)$, $\chi_{840}(407,·)$, $\chi_{840}(797,·)$, $\chi_{840}(799,·)$, $\chi_{840}(547,·)$, $\chi_{840}(293,·)$, $\chi_{840}(41,·)$, $\chi_{840}(811,·)$, $\chi_{840}(29,·)$, $\chi_{840}(433,·)$, $\chi_{840}(181,·)$, $\chi_{840}(671,·)$, $\chi_{840}(701,·)$, $\chi_{840}(839,·)$, $\chi_{840}(587,·)$, $\chi_{840}(209,·)$, $\chi_{840}(83,·)$, $\chi_{840}(727,·)$, $\chi_{840}(169,·)$, $\chi_{840}(349,·)$, $\chi_{840}(223,·)$, $\chi_{840}(97,·)$, $\chi_{840}(743,·)$, $\chi_{840}(617,·)$, $\chi_{840}(491,·)$, $\chi_{840}(113,·)$, $\chi_{840}(757,·)$, $\chi_{840}(631,·)$, $\chi_{840}(253,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5} a^{20} + \frac{1}{5} a^{16} + \frac{1}{5} a^{12} + \frac{1}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{21} + \frac{1}{5} a^{17} + \frac{1}{5} a^{13} + \frac{1}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{22} + \frac{1}{5} a^{18} + \frac{1}{5} a^{14} + \frac{1}{5} a^{10} + \frac{1}{5} a^{6}$, $\frac{1}{25} a^{23} + \frac{1}{25} a^{19} - \frac{4}{25} a^{15} + \frac{11}{25} a^{11} - \frac{9}{25} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{24} + \frac{1}{25} a^{20} - \frac{4}{25} a^{16} + \frac{11}{25} a^{12} - \frac{9}{25} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{25} a^{25} + \frac{1}{25} a^{21} - \frac{4}{25} a^{17} + \frac{11}{25} a^{13} - \frac{9}{25} a^{9} - \frac{1}{5} a^{5}$, $\frac{1}{125} a^{26} - \frac{4}{125} a^{22} - \frac{59}{125} a^{18} + \frac{6}{125} a^{14} + \frac{36}{125} a^{10} + \frac{8}{25} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{125} a^{27} + \frac{1}{125} a^{23} - \frac{54}{125} a^{19} - \frac{14}{125} a^{15} - \frac{34}{125} a^{11} - \frac{1}{25} a^{7}$, $\frac{1}{2242516497189755429938625} a^{28} - \frac{13230029602328663029754}{2242516497189755429938625} a^{24} + \frac{180099419666891406042816}{2242516497189755429938625} a^{20} - \frac{371345557514255928091369}{2242516497189755429938625} a^{16} - \frac{1052180002458774395496964}{2242516497189755429938625} a^{12} - \frac{9352958749349083509368}{23605436812523741367775} a^{8} + \frac{3395842044159075505211}{89700659887590217197545} a^{4} - \frac{7813241897297668447422}{17940131977518043439509}$, $\frac{1}{11212582485948777149693125} a^{29} + \frac{76470630285261554167791}{11212582485948777149693125} a^{25} - \frac{178703219883469462747364}{11212582485948777149693125} a^{21} - \frac{1178651496502567882869274}{11212582485948777149693125} a^{17} - \frac{2756492540322988522250319}{11212582485948777149693125} a^{13} - \frac{22572003364362378675322}{118027184062618706838875} a^{9} - \frac{32484421910877011373807}{448503299437951085987725} a^{5} - \frac{25753373874815711886931}{89700659887590217197545} a$, $\frac{1}{11212582485948777149693125} a^{30} - \frac{13230029602328663029754}{11212582485948777149693125} a^{26} + \frac{180099419666891406042816}{11212582485948777149693125} a^{22} + \frac{4113687436865254931785881}{11212582485948777149693125} a^{18} - \frac{3294696499648529825435589}{11212582485948777149693125} a^{14} - \frac{56563832374396566244918}{118027184062618706838875} a^{10} - \frac{176005477731021358889879}{448503299437951085987725} a^{6} - \frac{8738701170466751065288}{17940131977518043439509} a^{2}$, $\frac{1}{11212582485948777149693125} a^{31} - \frac{13230029602328663029754}{11212582485948777149693125} a^{27} + \frac{180099419666891406042816}{11212582485948777149693125} a^{23} + \frac{4113687436865254931785881}{11212582485948777149693125} a^{19} - \frac{3294696499648529825435589}{11212582485948777149693125} a^{15} - \frac{56563832374396566244918}{118027184062618706838875} a^{11} - \frac{176005477731021358889879}{448503299437951085987725} a^{7} - \frac{8738701170466751065288}{17940131977518043439509} a^{3}$
Class group and class number
$C_{4}\times C_{4}\times C_{4}\times C_{160}$, which has order $10240$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{2198369948712473}{37183285256952525625} a^{30} + \frac{518844669500441928}{37183285256952525625} a^{26} + \frac{23103012220152817688}{37183285256952525625} a^{22} + \frac{303127084902514156708}{37183285256952525625} a^{18} + \frac{1524857515813747103448}{37183285256952525625} a^{14} + \frac{663086681384178716988}{7436657051390505125} a^{10} + \frac{23965094275504764408}{297466282055620205} a^{6} + \frac{7485450216817807444}{297466282055620205} a^{2} \) (order $4$) | 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: | \( 2194214366954.93 \) (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 | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 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 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $7$ | 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 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |