Normalized defining polynomial
\( x^{32} + 31x^{24} + 705x^{16} + 7936x^{8} + 65536 \)
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: | \(113341328678310292663197722067951895712000278265856\) \(\medspace = 2^{96}\cdot 3^{16}\cdot 7^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(36.66\) | 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: | $2^{3}3^{1/2}7^{1/2}\approx 36.66060555964672$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(336=2^{4}\cdot 3\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{336}(1,·)$, $\chi_{336}(265,·)$, $\chi_{336}(139,·)$, $\chi_{336}(13,·)$, $\chi_{336}(281,·)$, $\chi_{336}(155,·)$, $\chi_{336}(29,·)$, $\chi_{336}(293,·)$, $\chi_{336}(167,·)$, $\chi_{336}(41,·)$, $\chi_{336}(43,·)$, $\chi_{336}(125,·)$, $\chi_{336}(307,·)$, $\chi_{336}(181,·)$, $\chi_{336}(55,·)$, $\chi_{336}(323,·)$, $\chi_{336}(197,·)$, $\chi_{336}(71,·)$, $\chi_{336}(335,·)$, $\chi_{336}(209,·)$, $\chi_{336}(83,·)$, $\chi_{336}(85,·)$, $\chi_{336}(223,·)$, $\chi_{336}(97,·)$, $\chi_{336}(295,·)$, $\chi_{336}(239,·)$, $\chi_{336}(113,·)$, $\chi_{336}(211,·)$, $\chi_{336}(169,·)$, $\chi_{336}(251,·)$, $\chi_{336}(253,·)$, $\chi_{336}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{8}+\frac{1}{3}$, $\frac{1}{6}a^{17}-\frac{1}{6}a^{9}+\frac{1}{6}a$, $\frac{1}{12}a^{18}-\frac{1}{12}a^{10}+\frac{1}{12}a^{2}$, $\frac{1}{24}a^{19}-\frac{1}{24}a^{11}+\frac{1}{24}a^{3}$, $\frac{1}{48}a^{20}-\frac{1}{48}a^{12}+\frac{1}{48}a^{4}$, $\frac{1}{96}a^{21}-\frac{1}{96}a^{13}+\frac{1}{96}a^{5}$, $\frac{1}{192}a^{22}+\frac{95}{192}a^{14}+\frac{1}{192}a^{6}$, $\frac{1}{384}a^{23}-\frac{97}{384}a^{15}-\frac{191}{384}a^{7}$, $\frac{1}{541440}a^{24}-\frac{11}{256}a^{16}-\frac{85}{256}a^{8}+\frac{736}{2115}$, $\frac{1}{1082880}a^{25}-\frac{11}{512}a^{17}-\frac{85}{512}a^{9}+\frac{368}{2115}a$, $\frac{1}{2165760}a^{26}-\frac{11}{1024}a^{18}-\frac{85}{1024}a^{10}+\frac{184}{2115}a^{2}$, $\frac{1}{4331520}a^{27}-\frac{11}{2048}a^{19}+\frac{939}{2048}a^{11}-\frac{1931}{4230}a^{3}$, $\frac{1}{8663040}a^{28}-\frac{11}{4096}a^{20}+\frac{939}{4096}a^{12}-\frac{1931}{8460}a^{4}$, $\frac{1}{17326080}a^{29}-\frac{11}{8192}a^{21}-\frac{3157}{8192}a^{13}+\frac{6529}{16920}a^{5}$, $\frac{1}{34652160}a^{30}-\frac{11}{16384}a^{22}+\frac{5035}{16384}a^{14}+\frac{6529}{33840}a^{6}$, $\frac{1}{69304320}a^{31}-\frac{11}{32768}a^{23}+\frac{5035}{32768}a^{15}+\frac{6529}{67680}a^{7}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{4}\times C_{4}\times C_{4}$, which has order $64$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{7}{270720} a^{31} + \frac{7193}{270720} a^{7} \) (order $48$) | 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: | $\frac{209}{2887680}a^{28}+\frac{15}{4096}a^{20}+\frac{209}{4096}a^{12}+\frac{6479}{11280}a^{4}+1$, $\frac{457}{4331520}a^{27}-\frac{271}{1082880}a^{25}+\frac{23}{6144}a^{19}-\frac{17}{1536}a^{17}+\frac{457}{6144}a^{11}-\frac{271}{1536}a^{9}+\frac{14167}{16920}a^{3}-\frac{8401}{4230}a$, $\frac{1}{27072}a^{30}+\frac{209}{2887680}a^{28}-\frac{341}{2165760}a^{26}+\frac{15}{4096}a^{20}-\frac{11}{3072}a^{18}+\frac{209}{4096}a^{12}-\frac{85}{3072}a^{10}-\frac{4895}{27072}a^{6}+\frac{6479}{11280}a^{4}-\frac{704}{2115}a^{2}+1$, $\frac{1541}{34652160}a^{30}-\frac{287}{17326080}a^{29}+\frac{91}{49152}a^{22}-\frac{1}{24576}a^{21}+\frac{1541}{49152}a^{14}-\frac{287}{24576}a^{13}+\frac{47771}{135360}a^{6}-\frac{8897}{67680}a^{5}$, $\frac{271}{1082880}a^{25}+\frac{17}{1536}a^{17}+\frac{271}{1536}a^{9}+\frac{8401}{4230}a+1$, $\frac{457}{4331520}a^{27}+\frac{23}{6144}a^{19}+\frac{457}{6144}a^{11}+\frac{14167}{16920}a^{3}+1$, $\frac{1}{27072}a^{30}-\frac{271}{1082880}a^{25}-\frac{17}{1536}a^{17}-\frac{271}{1536}a^{9}-\frac{4895}{27072}a^{6}-\frac{8401}{4230}a$, $\frac{43}{2165760}a^{27}-\frac{11}{3072}a^{19}-\frac{85}{3072}a^{11}-\frac{13549}{16920}a^{3}$, $\frac{967}{69304320}a^{31}+\frac{209}{2887680}a^{28}+\frac{713}{4331520}a^{27}+\frac{1}{8460}a^{26}-\frac{31}{180480}a^{25}-\frac{31}{180480}a^{24}+\frac{89}{98304}a^{23}+\frac{15}{4096}a^{20}+\frac{23}{6144}a^{19}-\frac{1}{256}a^{17}-\frac{1}{256}a^{16}+\frac{967}{98304}a^{15}+\frac{209}{4096}a^{12}+\frac{457}{6144}a^{11}-\frac{31}{256}a^{9}-\frac{31}{256}a^{8}+\frac{29977}{270720}a^{7}+\frac{6479}{11280}a^{4}+\frac{736}{2115}a^{3}-\frac{8279}{8460}a^{2}-\frac{961}{705}a-\frac{256}{705}$, $\frac{1}{11280}a^{28}-\frac{17}{433152}a^{27}+\frac{713}{2165760}a^{26}+\frac{1}{4230}a^{25}-\frac{11}{3072}a^{19}+\frac{23}{3072}a^{18}-\frac{85}{3072}a^{11}+\frac{457}{3072}a^{10}-\frac{2639}{11280}a^{4}-\frac{527}{1692}a^{3}+\frac{1472}{2115}a^{2}-\frac{4049}{4230}a$, $\frac{1}{32768}a^{31}+\frac{1}{67680}a^{29}+\frac{209}{2887680}a^{28}-\frac{1}{4230}a^{26}-\frac{271}{1082880}a^{25}-\frac{31}{180480}a^{24}+\frac{31}{32768}a^{23}+\frac{15}{4096}a^{20}-\frac{17}{1536}a^{17}-\frac{1}{256}a^{16}+\frac{705}{32768}a^{15}+\frac{209}{4096}a^{12}-\frac{271}{1536}a^{9}-\frac{31}{256}a^{8}+\frac{31}{128}a^{7}+\frac{8641}{67680}a^{5}+\frac{6479}{11280}a^{4}+\frac{4049}{4230}a^{2}-\frac{8401}{4230}a-\frac{256}{705}$, $\frac{967}{69304320}a^{31}-\frac{1}{11280}a^{29}+\frac{1}{8460}a^{26}+\frac{31}{180480}a^{24}+\frac{89}{98304}a^{23}+\frac{1}{256}a^{16}+\frac{967}{98304}a^{15}+\frac{31}{256}a^{8}+\frac{29977}{270720}a^{7}+\frac{2639}{11280}a^{5}-\frac{8279}{8460}a^{2}+\frac{256}{705}$, $\frac{7}{270720}a^{31}-\frac{1}{67680}a^{29}-\frac{17}{433152}a^{27}+\frac{17}{216576}a^{25}-\frac{11}{3072}a^{19}+\frac{11}{1536}a^{17}-\frac{85}{3072}a^{11}+\frac{85}{1536}a^{9}-\frac{7193}{270720}a^{7}-\frac{8641}{67680}a^{5}-\frac{527}{1692}a^{3}+\frac{527}{846}a$, $\frac{961}{23101440}a^{31}-\frac{1}{27072}a^{30}-\frac{1}{11280}a^{28}-\frac{17}{433152}a^{27}+\frac{1}{8460}a^{26}-\frac{31}{180480}a^{24}+\frac{31}{32768}a^{23}-\frac{11}{3072}a^{19}-\frac{1}{256}a^{16}+\frac{705}{32768}a^{15}-\frac{85}{3072}a^{11}-\frac{31}{256}a^{8}+\frac{62}{705}a^{7}+\frac{4895}{27072}a^{6}+\frac{2639}{11280}a^{4}-\frac{527}{1692}a^{3}+\frac{181}{8460}a^{2}-\frac{256}{705}$, $\frac{2821}{34652160}a^{30}+\frac{1541}{17326080}a^{29}-\frac{1}{11280}a^{28}+\frac{341}{2165760}a^{26}-\frac{31}{180480}a^{25}+\frac{91}{49152}a^{22}+\frac{91}{24576}a^{21}+\frac{11}{3072}a^{18}-\frac{1}{256}a^{17}+\frac{1541}{49152}a^{14}+\frac{1541}{24576}a^{13}+\frac{85}{3072}a^{10}-\frac{31}{256}a^{9}+\frac{364}{2115}a^{6}+\frac{47771}{67680}a^{5}+\frac{2639}{11280}a^{4}+\frac{704}{2115}a^{2}-\frac{961}{705}a-1$ (assuming GRH) | 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: | \( 374180447387.275 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 374180447387.275 \cdot 64}{48\cdot\sqrt{113341328678310292663197722067951895712000278265856}}\cr\approx \mathstrut & 0.276505370236942 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/5.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{16}$ | ${\href{/padicField/19.4.0.1}{4} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{16}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{16}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.48.1 | $x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$ | $8$ | $2$ | $48$ | $C_4\times C_2^2$ | $[2, 3, 4]^{2}$ |
2.16.48.1 | $x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$ | $8$ | $2$ | $48$ | $C_4\times C_2^2$ | $[2, 3, 4]^{2}$ | |
\(3\) | 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |