Normalized defining polynomial
\( x^{32} + 13 x^{28} + 78 x^{24} + 356 x^{20} + 1505 x^{16} + 5696 x^{12} + 19968 x^{8} + 53248 x^{4} + 65536 \)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(8779518136340480467664896000000000000000000000000\)\(\medspace = 2^{64}\cdot 5^{24}\cdot 41^{8}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $33.84$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 5, 41$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $16$ | ||
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{15} a^{16} - \frac{1}{15} a^{12} + \frac{1}{15} a^{8} - \frac{1}{15} a^{4} + \frac{1}{15}$, $\frac{1}{30} a^{17} - \frac{1}{30} a^{13} - \frac{7}{15} a^{9} + \frac{7}{15} a^{5} + \frac{1}{30} a$, $\frac{1}{60} a^{18} + \frac{29}{60} a^{14} - \frac{7}{30} a^{10} - \frac{4}{15} a^{6} + \frac{1}{60} a^{2}$, $\frac{1}{120} a^{19} + \frac{29}{120} a^{15} + \frac{23}{60} a^{11} + \frac{11}{30} a^{7} + \frac{1}{120} a^{3}$, $\frac{1}{240} a^{20} - \frac{1}{80} a^{16} + \frac{13}{40} a^{12} - \frac{9}{20} a^{8} + \frac{11}{80} a^{4} - \frac{2}{15}$, $\frac{1}{480} a^{21} - \frac{1}{160} a^{17} - \frac{27}{80} a^{13} + \frac{11}{40} a^{9} + \frac{11}{160} a^{5} + \frac{13}{30} a$, $\frac{1}{960} a^{22} - \frac{1}{320} a^{18} + \frac{53}{160} a^{14} + \frac{11}{80} a^{10} + \frac{11}{320} a^{6} - \frac{17}{60} a^{2}$, $\frac{1}{1920} a^{23} - \frac{1}{640} a^{19} - \frac{107}{320} a^{15} + \frac{11}{160} a^{11} + \frac{11}{640} a^{7} + \frac{43}{120} a^{3}$, $\frac{1}{3840} a^{24} - \frac{1}{1280} a^{20} + \frac{21}{640} a^{16} - \frac{53}{320} a^{12} - \frac{373}{1280} a^{8} + \frac{23}{48} a^{4} + \frac{1}{5}$, $\frac{1}{7680} a^{25} - \frac{1}{2560} a^{21} + \frac{21}{1280} a^{17} - \frac{53}{640} a^{13} - \frac{373}{2560} a^{9} - \frac{25}{96} a^{5} + \frac{1}{10} a$, $\frac{1}{15360} a^{26} - \frac{1}{5120} a^{22} + \frac{21}{2560} a^{18} - \frac{53}{1280} a^{14} + \frac{2187}{5120} a^{10} - \frac{25}{192} a^{6} + \frac{1}{20} a^{2}$, $\frac{1}{30720} a^{27} - \frac{1}{10240} a^{23} + \frac{21}{5120} a^{19} + \frac{1227}{2560} a^{15} - \frac{2933}{10240} a^{11} + \frac{167}{384} a^{7} + \frac{1}{40} a^{3}$, $\frac{1}{5836800} a^{28} + \frac{397}{5836800} a^{24} + \frac{1101}{972800} a^{20} - \frac{6907}{291840} a^{16} - \frac{511187}{1167360} a^{12} - \frac{4801}{91200} a^{8} - \frac{829}{1900} a^{4} - \frac{479}{1425}$, $\frac{1}{11673600} a^{29} + \frac{397}{11673600} a^{25} + \frac{1101}{1945600} a^{21} - \frac{6907}{583680} a^{17} + \frac{656173}{2334720} a^{13} + \frac{86399}{182400} a^{9} + \frac{1071}{3800} a^{5} - \frac{479}{2850} a$, $\frac{1}{23347200} a^{30} + \frac{397}{23347200} a^{26} + \frac{1101}{3891200} a^{22} - \frac{6907}{1167360} a^{18} + \frac{656173}{4669440} a^{14} - \frac{96001}{364800} a^{10} + \frac{1071}{7600} a^{6} + \frac{2371}{5700} a^{2}$, $\frac{1}{46694400} a^{31} + \frac{397}{46694400} a^{27} + \frac{1101}{7782400} a^{23} - \frac{6907}{2334720} a^{19} + \frac{656173}{9338880} a^{15} + \frac{268799}{729600} a^{11} - \frac{6529}{15200} a^{7} - \frac{3329}{11400} a^{3}$
Class group and class number
$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (assuming GRH)
Unit group
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -\frac{469}{5836800} a^{31} - \frac{4553}{5836800} a^{27} - \frac{9727}{2918400} a^{23} - \frac{4829}{291840} a^{19} - \frac{78289}{1167360} a^{15} - \frac{170503}{729600} a^{11} - \frac{37391}{45600} a^{7} - \frac{5669}{3800} a^{3} \) (order $40$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 150821321458.18454 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2^2\times C_2^2:C_4$ (as 32T262):
| A solvable group of order 64 |
| The 40 conjugacy class representatives for $C_2^2\times C_2^2:C_4$ |
| Character table for $C_2^2\times C_2^2:C_4$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 32 siblings: | data not computed |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$ |
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 | ||||||
| $5$ | 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 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $41$ | 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |