Normalized defining polynomial
\( x^{32} + 2 x^{30} - 56 x^{28} - 824 x^{26} + 992 x^{24} + 23552 x^{22} + 121088 x^{20} + 560896 x^{18} + \cdots + 65536 \)
Invariants
Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(361280028614042283247927296000000000000000000000000\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 29^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.01\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{3/2}3^{1/2}5^{3/4}29^{1/2}\approx 88.21290473450735$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{768}a^{16}-\frac{1}{384}a^{14}+\frac{1}{96}a^{10}-\frac{1}{48}a^{8}+\frac{1}{24}a^{6}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{768}a^{17}-\frac{1}{384}a^{15}+\frac{1}{96}a^{11}-\frac{1}{48}a^{9}+\frac{1}{24}a^{7}-\frac{1}{6}a^{3}+\frac{1}{3}a$, $\frac{1}{1536}a^{18}-\frac{1}{384}a^{14}+\frac{1}{192}a^{12}+\frac{1}{24}a^{6}-\frac{1}{12}a^{4}+\frac{1}{3}$, $\frac{1}{1536}a^{19}-\frac{1}{384}a^{15}+\frac{1}{192}a^{13}+\frac{1}{24}a^{7}-\frac{1}{12}a^{5}+\frac{1}{3}a$, $\frac{1}{12288}a^{20}-\frac{5}{384}a^{10}+\frac{1}{12}$, $\frac{1}{12288}a^{21}-\frac{5}{384}a^{11}+\frac{1}{12}a$, $\frac{1}{24576}a^{22}-\frac{5}{768}a^{12}+\frac{1}{24}a^{2}$, $\frac{1}{24576}a^{23}-\frac{5}{768}a^{13}+\frac{1}{24}a^{3}$, $\frac{1}{5898240}a^{24}+\frac{1}{196608}a^{22}+\frac{1}{32768}a^{20}-\frac{41}{184320}a^{18}+\frac{1}{6144}a^{16}-\frac{11}{12288}a^{14}-\frac{419}{92160}a^{12}-\frac{35}{3072}a^{10}-\frac{7}{384}a^{8}-\frac{161}{2880}a^{6}+\frac{11}{384}a^{4}+\frac{11}{64}a^{2}-\frac{599}{1440}$, $\frac{1}{5898240}a^{25}+\frac{1}{196608}a^{23}+\frac{1}{32768}a^{21}-\frac{41}{184320}a^{19}+\frac{1}{6144}a^{17}-\frac{11}{12288}a^{15}-\frac{419}{92160}a^{13}-\frac{35}{3072}a^{11}-\frac{7}{384}a^{9}-\frac{161}{2880}a^{7}+\frac{11}{384}a^{5}+\frac{11}{64}a^{3}-\frac{599}{1440}a$, $\frac{1}{342097920}a^{26}-\frac{1}{28508160}a^{24}+\frac{35}{2850816}a^{22}-\frac{389}{42762240}a^{20}+\frac{21}{148480}a^{18}-\frac{53}{237568}a^{16}-\frac{3517}{2672640}a^{14}-\frac{761}{445440}a^{12}-\frac{653}{89088}a^{10}+\frac{391}{41760}a^{8}+\frac{881}{37120}a^{6}-\frac{35}{5568}a^{4}-\frac{5797}{41760}a^{2}+\frac{2513}{13920}$, $\frac{1}{342097920}a^{27}-\frac{1}{28508160}a^{25}+\frac{35}{2850816}a^{23}-\frac{389}{42762240}a^{21}+\frac{21}{148480}a^{19}-\frac{53}{237568}a^{17}-\frac{3517}{2672640}a^{15}-\frac{761}{445440}a^{13}-\frac{653}{89088}a^{11}+\frac{391}{41760}a^{9}+\frac{881}{37120}a^{7}-\frac{35}{5568}a^{5}-\frac{5797}{41760}a^{3}+\frac{2513}{13920}a$, $\frac{1}{3067934146560}a^{28}+\frac{61}{51132235776}a^{26}+\frac{21589}{255661178880}a^{24}-\frac{2103697}{191745884160}a^{22}+\frac{29465}{4261019648}a^{20}+\frac{4520329}{31957647360}a^{18}+\frac{13057283}{23968235520}a^{16}+\frac{1966453}{532627456}a^{14}+\frac{1974939}{665784320}a^{12}+\frac{79457851}{5992058880}a^{10}+\frac{1500217}{66578432}a^{8}-\frac{10249163}{249669120}a^{6}+\frac{70804351}{749007360}a^{4}-\frac{233107}{12483456}a^{2}+\frac{6014539}{62417280}$, $\frac{1}{3067934146560}a^{29}+\frac{61}{51132235776}a^{27}+\frac{21589}{255661178880}a^{25}-\frac{2103697}{191745884160}a^{23}+\frac{29465}{4261019648}a^{21}+\frac{4520329}{31957647360}a^{19}+\frac{13057283}{23968235520}a^{17}+\frac{1966453}{532627456}a^{15}+\frac{1974939}{665784320}a^{13}+\frac{79457851}{5992058880}a^{11}+\frac{1500217}{66578432}a^{9}-\frac{10249163}{249669120}a^{7}+\frac{70804351}{749007360}a^{5}-\frac{233107}{12483456}a^{3}+\frac{6014539}{62417280}a$, $\frac{1}{951059585433600}a^{30}+\frac{1}{7669835366400}a^{28}+\frac{10907}{26418321817600}a^{26}+\frac{289063}{11888244817920}a^{24}+\frac{834797477}{59441224089600}a^{22}+\frac{3838369}{341616230400}a^{20}+\frac{78744139}{1486030602240}a^{18}-\frac{3148591843}{7430153011200}a^{16}-\frac{131615707}{619179417600}a^{14}+\frac{2496327887}{371507650560}a^{12}-\frac{4970558263}{928769126400}a^{10}-\frac{9887309}{25799142400}a^{8}-\frac{996093481}{46438456320}a^{6}+\frac{4077201151}{58048070400}a^{4}+\frac{31126021}{624172800}a^{2}-\frac{1088448923}{2418669600}$, $\frac{1}{951059585433600}a^{31}+\frac{1}{7669835366400}a^{29}+\frac{10907}{26418321817600}a^{27}+\frac{289063}{11888244817920}a^{25}+\frac{834797477}{59441224089600}a^{23}+\frac{3838369}{341616230400}a^{21}+\frac{78744139}{1486030602240}a^{19}-\frac{3148591843}{7430153011200}a^{17}-\frac{131615707}{619179417600}a^{15}+\frac{2496327887}{371507650560}a^{13}-\frac{4970558263}{928769126400}a^{11}-\frac{9887309}{25799142400}a^{9}-\frac{996093481}{46438456320}a^{7}+\frac{4077201151}{58048070400}a^{5}+\frac{31126021}{624172800}a^{3}-\frac{1088448923}{2418669600}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
not computed
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{703968131}{475529792716800} a^{30} - \frac{2058181}{807351091200} a^{28} + \frac{2449937513}{29720612044800} a^{26} + \frac{28367837293}{23776489635840} a^{24} - \frac{25802783791}{14860306022400} a^{22} - \frac{52135855529}{1564242739200} a^{20} - \frac{505243473911}{2972061204480} a^{18} - \frac{1509983544481}{1857538252800} a^{16} - \frac{9825907907561}{3715076505600} a^{14} - \frac{158224176703}{46438456320} a^{12} - \frac{4465453693909}{928769126400} a^{10} - \frac{2713845539621}{464384563200} a^{8} - \frac{9376358513}{2902403520} a^{6} - \frac{199396520239}{116096140800} a^{4} - \frac{536714549}{468129600} a^{2} + \frac{6195366329}{9674678400} \) (order $30$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
$C_2^4:C_4$ (as 32T262):
A solvable group of order 64 |
The 40 conjugacy class representatives for $C_2^4:C_4$ |
Character table for $C_2^4:C_4$ |
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 |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{16}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(29\) | 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |