Normalized defining polynomial
\( x^{16} + 24 x^{14} - 24 x^{13} + 388 x^{12} - 552 x^{11} + 4236 x^{10} - 7104 x^{9} + 32370 x^{8} + \cdots + 855742 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(22144295318673432094540038144\) \(\medspace = 2^{48}\cdot 3^{12}\cdot 23^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(59.10\) | 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: | $2^{3}3^{3/4}23^{1/2}\approx 87.45705441086021$ | ||
Ramified primes: | \(2\), \(3\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{143}a^{12}-\frac{3}{11}a^{11}+\frac{34}{143}a^{10}-\frac{10}{143}a^{9}-\frac{1}{13}a^{8}-\frac{7}{143}a^{7}+\frac{1}{13}a^{6}-\frac{42}{143}a^{5}+\frac{68}{143}a^{4}+\frac{2}{11}a^{3}+\frac{7}{143}a^{2}-\frac{47}{143}a-\frac{28}{143}$, $\frac{1}{143}a^{13}-\frac{57}{143}a^{11}+\frac{29}{143}a^{10}+\frac{28}{143}a^{9}-\frac{7}{143}a^{8}+\frac{24}{143}a^{7}-\frac{42}{143}a^{6}+\frac{3}{143}a^{5}-\frac{3}{11}a^{4}+\frac{20}{143}a^{3}-\frac{60}{143}a^{2}-\frac{2}{143}a+\frac{4}{11}$, $\frac{1}{2746028857}a^{14}+\frac{8619222}{2746028857}a^{13}+\frac{2228507}{2746028857}a^{12}-\frac{19503288}{249638987}a^{11}-\frac{80807488}{2746028857}a^{10}-\frac{646776538}{2746028857}a^{9}+\frac{769440273}{2746028857}a^{8}-\frac{446402260}{2746028857}a^{7}+\frac{1558230}{9184043}a^{6}+\frac{1196009571}{2746028857}a^{5}-\frac{131059935}{2746028857}a^{4}+\frac{1365665534}{2746028857}a^{3}+\frac{953625065}{2746028857}a^{2}-\frac{1364536214}{2746028857}a+\frac{84848416}{211232989}$, $\frac{1}{17\!\cdots\!11}a^{15}+\frac{23\!\cdots\!98}{17\!\cdots\!11}a^{14}-\frac{15\!\cdots\!52}{17\!\cdots\!11}a^{13}-\frac{84\!\cdots\!93}{17\!\cdots\!11}a^{12}+\frac{55\!\cdots\!49}{17\!\cdots\!11}a^{11}+\frac{25\!\cdots\!11}{16\!\cdots\!01}a^{10}+\frac{51\!\cdots\!23}{17\!\cdots\!11}a^{9}-\frac{15\!\cdots\!03}{77\!\cdots\!57}a^{8}+\frac{68\!\cdots\!88}{17\!\cdots\!11}a^{7}-\frac{51\!\cdots\!21}{17\!\cdots\!11}a^{6}+\frac{78\!\cdots\!04}{17\!\cdots\!11}a^{5}+\frac{62\!\cdots\!19}{13\!\cdots\!47}a^{4}-\frac{22\!\cdots\!67}{17\!\cdots\!11}a^{3}+\frac{52\!\cdots\!65}{17\!\cdots\!11}a^{2}-\frac{74\!\cdots\!72}{17\!\cdots\!11}a+\frac{87\!\cdots\!62}{17\!\cdots\!11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{8}$, which has order $256$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{48\!\cdots\!98}{10\!\cdots\!03}a^{15}-\frac{53\!\cdots\!63}{10\!\cdots\!03}a^{14}+\frac{10\!\cdots\!32}{10\!\cdots\!03}a^{13}-\frac{21\!\cdots\!00}{10\!\cdots\!03}a^{12}+\frac{15\!\cdots\!24}{98\!\cdots\!73}a^{11}-\frac{34\!\cdots\!99}{98\!\cdots\!73}a^{10}+\frac{17\!\cdots\!24}{10\!\cdots\!03}a^{9}-\frac{40\!\cdots\!17}{10\!\cdots\!03}a^{8}+\frac{12\!\cdots\!24}{10\!\cdots\!03}a^{7}-\frac{34\!\cdots\!80}{10\!\cdots\!03}a^{6}+\frac{43\!\cdots\!00}{75\!\cdots\!21}a^{5}-\frac{13\!\cdots\!21}{10\!\cdots\!03}a^{4}+\frac{15\!\cdots\!48}{10\!\cdots\!03}a^{3}+\frac{30\!\cdots\!56}{10\!\cdots\!03}a^{2}+\frac{57\!\cdots\!66}{98\!\cdots\!73}a+\frac{20\!\cdots\!39}{10\!\cdots\!03}$, $\frac{48\!\cdots\!88}{11\!\cdots\!21}a^{15}+\frac{68\!\cdots\!00}{12\!\cdots\!31}a^{14}+\frac{11\!\cdots\!88}{12\!\cdots\!31}a^{13}-\frac{29\!\cdots\!71}{99\!\cdots\!87}a^{12}+\frac{15\!\cdots\!48}{12\!\cdots\!31}a^{11}-\frac{12\!\cdots\!24}{12\!\cdots\!31}a^{10}+\frac{14\!\cdots\!90}{12\!\cdots\!31}a^{9}-\frac{20\!\cdots\!03}{12\!\cdots\!31}a^{8}+\frac{97\!\cdots\!84}{12\!\cdots\!31}a^{7}-\frac{24\!\cdots\!82}{12\!\cdots\!31}a^{6}+\frac{23\!\cdots\!60}{12\!\cdots\!31}a^{5}-\frac{10\!\cdots\!92}{12\!\cdots\!31}a^{4}+\frac{12\!\cdots\!72}{12\!\cdots\!31}a^{3}+\frac{11\!\cdots\!02}{12\!\cdots\!31}a^{2}+\frac{80\!\cdots\!72}{12\!\cdots\!31}a+\frac{48\!\cdots\!17}{11\!\cdots\!21}$, $\frac{11\!\cdots\!40}{98\!\cdots\!73}a^{15}+\frac{65\!\cdots\!44}{83\!\cdots\!31}a^{14}+\frac{27\!\cdots\!60}{10\!\cdots\!03}a^{13}-\frac{18\!\cdots\!73}{10\!\cdots\!03}a^{12}+\frac{38\!\cdots\!80}{10\!\cdots\!03}a^{11}-\frac{36\!\cdots\!94}{83\!\cdots\!31}a^{10}+\frac{38\!\cdots\!26}{10\!\cdots\!03}a^{9}-\frac{60\!\cdots\!76}{10\!\cdots\!03}a^{8}+\frac{24\!\cdots\!24}{10\!\cdots\!03}a^{7}-\frac{65\!\cdots\!70}{10\!\cdots\!03}a^{6}+\frac{86\!\cdots\!64}{10\!\cdots\!03}a^{5}-\frac{26\!\cdots\!88}{10\!\cdots\!03}a^{4}+\frac{21\!\cdots\!28}{83\!\cdots\!31}a^{3}+\frac{26\!\cdots\!98}{10\!\cdots\!03}a^{2}+\frac{19\!\cdots\!28}{10\!\cdots\!03}a+\frac{30\!\cdots\!23}{98\!\cdots\!73}$, $\frac{44\!\cdots\!49}{17\!\cdots\!11}a^{15}+\frac{46\!\cdots\!55}{17\!\cdots\!11}a^{14}+\frac{10\!\cdots\!12}{17\!\cdots\!11}a^{13}-\frac{64\!\cdots\!17}{13\!\cdots\!47}a^{12}+\frac{15\!\cdots\!16}{17\!\cdots\!11}a^{11}-\frac{19\!\cdots\!81}{17\!\cdots\!11}a^{10}+\frac{16\!\cdots\!53}{17\!\cdots\!11}a^{9}-\frac{23\!\cdots\!72}{17\!\cdots\!11}a^{8}+\frac{10\!\cdots\!78}{16\!\cdots\!01}a^{7}-\frac{24\!\cdots\!17}{17\!\cdots\!11}a^{6}+\frac{48\!\cdots\!49}{17\!\cdots\!11}a^{5}-\frac{11\!\cdots\!56}{17\!\cdots\!11}a^{4}+\frac{97\!\cdots\!48}{17\!\cdots\!11}a^{3}-\frac{95\!\cdots\!19}{17\!\cdots\!11}a^{2}-\frac{70\!\cdots\!70}{13\!\cdots\!47}a+\frac{77\!\cdots\!65}{13\!\cdots\!47}$, $\frac{14\!\cdots\!22}{17\!\cdots\!11}a^{15}+\frac{13\!\cdots\!51}{17\!\cdots\!11}a^{14}+\frac{72\!\cdots\!71}{17\!\cdots\!11}a^{13}+\frac{23\!\cdots\!38}{16\!\cdots\!01}a^{12}+\frac{12\!\cdots\!16}{17\!\cdots\!11}a^{11}+\frac{32\!\cdots\!14}{17\!\cdots\!11}a^{10}+\frac{14\!\cdots\!20}{17\!\cdots\!11}a^{9}+\frac{19\!\cdots\!89}{17\!\cdots\!11}a^{8}+\frac{12\!\cdots\!48}{17\!\cdots\!11}a^{7}+\frac{38\!\cdots\!12}{17\!\cdots\!11}a^{6}+\frac{32\!\cdots\!51}{17\!\cdots\!11}a^{5}-\frac{68\!\cdots\!90}{13\!\cdots\!47}a^{4}-\frac{23\!\cdots\!24}{17\!\cdots\!11}a^{3}-\frac{77\!\cdots\!49}{17\!\cdots\!11}a^{2}-\frac{59\!\cdots\!24}{16\!\cdots\!01}a+\frac{31\!\cdots\!03}{17\!\cdots\!11}$, $\frac{66\!\cdots\!35}{17\!\cdots\!11}a^{15}+\frac{59\!\cdots\!55}{17\!\cdots\!11}a^{14}+\frac{12\!\cdots\!29}{17\!\cdots\!11}a^{13}+\frac{74\!\cdots\!13}{17\!\cdots\!11}a^{12}+\frac{27\!\cdots\!44}{17\!\cdots\!11}a^{11}+\frac{10\!\cdots\!27}{17\!\cdots\!11}a^{10}-\frac{57\!\cdots\!85}{17\!\cdots\!11}a^{9}+\frac{78\!\cdots\!90}{17\!\cdots\!11}a^{8}-\frac{13\!\cdots\!40}{17\!\cdots\!11}a^{7}+\frac{29\!\cdots\!15}{17\!\cdots\!11}a^{6}-\frac{19\!\cdots\!73}{17\!\cdots\!11}a^{5}+\frac{12\!\cdots\!79}{17\!\cdots\!11}a^{4}-\frac{94\!\cdots\!52}{17\!\cdots\!11}a^{3}+\frac{96\!\cdots\!57}{17\!\cdots\!11}a^{2}+\frac{82\!\cdots\!62}{17\!\cdots\!11}a-\frac{40\!\cdots\!91}{17\!\cdots\!11}$, $\frac{21\!\cdots\!81}{17\!\cdots\!11}a^{15}+\frac{12\!\cdots\!65}{17\!\cdots\!11}a^{14}+\frac{48\!\cdots\!10}{17\!\cdots\!11}a^{13}-\frac{22\!\cdots\!46}{17\!\cdots\!11}a^{12}+\frac{74\!\cdots\!76}{17\!\cdots\!11}a^{11}-\frac{65\!\cdots\!94}{17\!\cdots\!11}a^{10}+\frac{75\!\cdots\!35}{17\!\cdots\!11}a^{9}-\frac{88\!\cdots\!78}{17\!\cdots\!11}a^{8}+\frac{51\!\cdots\!16}{17\!\cdots\!11}a^{7}-\frac{93\!\cdots\!35}{16\!\cdots\!01}a^{6}+\frac{19\!\cdots\!88}{17\!\cdots\!11}a^{5}-\frac{50\!\cdots\!04}{17\!\cdots\!11}a^{4}+\frac{30\!\cdots\!64}{13\!\cdots\!47}a^{3}-\frac{41\!\cdots\!24}{17\!\cdots\!11}a^{2}-\frac{39\!\cdots\!56}{17\!\cdots\!11}a+\frac{48\!\cdots\!79}{16\!\cdots\!01}$ (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: | \( 204141.742403 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 204141.742403 \cdot 256}{2\cdot\sqrt{22144295318673432094540038144}}\cr\approx \mathstrut & 0.426530214870 \end{aligned}\] (assuming GRH)
Galois group
$C_4^2:C_2$ (as 16T30):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4^2:C_2$ |
Character table for $C_4^2:C_2$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.4.105984.2, 4.4.105984.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.404373897216.34, 8.8.179721732096.1, 8.0.1617495588864.18 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 sibling: | deg 16 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.48.13 | $x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{6} + 8 x^{2} + 8 x + 14$ | $16$ | $1$ | $48$ | 16T30 | $[2, 3, 3, 7/2]^{2}$ |
\(3\) | 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |