Normalized defining polynomial
\( x^{16} - 5 x^{15} + 20 x^{14} - 53 x^{13} + 425 x^{12} - 1761 x^{11} + 8234 x^{10} - 26439 x^{9} + 85127 x^{8} - 227294 x^{7} + 576600 x^{6} - 1117344 x^{5} + \cdots + 5607424 \)
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)
| |
Discriminant: | \(1077123334824966013513439398801\) \(\medspace = 41^{14}\cdot 73^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(75.34\) | 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: | $41^{7/8}73^{1/2}\approx 220.21520635937404$ | ||
Ramified primes: | \(41\), \(73\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}+\frac{3}{8}a^{8}-\frac{3}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{11}+\frac{3}{16}a^{9}-\frac{3}{16}a^{8}+\frac{3}{16}a^{7}-\frac{1}{8}a^{6}-\frac{7}{16}a^{5}+\frac{3}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{5}{64}a^{10}-\frac{11}{64}a^{9}+\frac{3}{64}a^{8}+\frac{3}{32}a^{7}+\frac{1}{64}a^{6}+\frac{11}{64}a^{5}-\frac{1}{32}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{468224}a^{14}-\frac{673}{468224}a^{13}-\frac{167}{14632}a^{12}+\frac{197}{15104}a^{11}+\frac{17077}{468224}a^{10}+\frac{24099}{468224}a^{9}+\frac{15491}{234112}a^{8}-\frac{199583}{468224}a^{7}+\frac{96395}{468224}a^{6}+\frac{116911}{234112}a^{5}+\frac{6463}{14632}a^{4}-\frac{655}{3658}a^{3}-\frac{5793}{29264}a^{2}+\frac{2559}{7316}a+\frac{564}{1829}$, $\frac{1}{14\!\cdots\!28}a^{15}-\frac{77\!\cdots\!77}{14\!\cdots\!28}a^{14}+\frac{59\!\cdots\!97}{37\!\cdots\!32}a^{13}-\frac{53\!\cdots\!17}{14\!\cdots\!28}a^{12}+\frac{86\!\cdots\!45}{14\!\cdots\!28}a^{11}+\frac{15\!\cdots\!43}{14\!\cdots\!28}a^{10}-\frac{13\!\cdots\!99}{74\!\cdots\!64}a^{9}-\frac{15\!\cdots\!91}{14\!\cdots\!28}a^{8}+\frac{63\!\cdots\!15}{14\!\cdots\!28}a^{7}+\frac{76\!\cdots\!85}{74\!\cdots\!64}a^{6}+\frac{16\!\cdots\!37}{18\!\cdots\!16}a^{5}+\frac{91\!\cdots\!89}{23\!\cdots\!52}a^{4}+\frac{12\!\cdots\!51}{92\!\cdots\!08}a^{3}+\frac{99\!\cdots\!93}{58\!\cdots\!88}a^{2}-\frac{21\!\cdots\!79}{58\!\cdots\!88}a+\frac{47\!\cdots\!60}{39\!\cdots\!31}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{33}$, which has order $33$ (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: | $\frac{78\!\cdots\!87}{35\!\cdots\!52}a^{15}-\frac{70\!\cdots\!01}{35\!\cdots\!52}a^{14}+\frac{12\!\cdots\!17}{17\!\cdots\!76}a^{13}-\frac{57\!\cdots\!51}{35\!\cdots\!52}a^{12}+\frac{30\!\cdots\!85}{35\!\cdots\!52}a^{11}-\frac{22\!\cdots\!61}{35\!\cdots\!52}a^{10}+\frac{23\!\cdots\!43}{89\!\cdots\!88}a^{9}-\frac{31\!\cdots\!01}{35\!\cdots\!52}a^{8}+\frac{77\!\cdots\!31}{35\!\cdots\!52}a^{7}-\frac{56\!\cdots\!03}{89\!\cdots\!88}a^{6}+\frac{12\!\cdots\!69}{89\!\cdots\!88}a^{5}-\frac{30\!\cdots\!71}{11\!\cdots\!36}a^{4}+\frac{76\!\cdots\!57}{22\!\cdots\!72}a^{3}-\frac{22\!\cdots\!43}{36\!\cdots\!56}a^{2}+\frac{25\!\cdots\!21}{28\!\cdots\!84}a-\frac{86\!\cdots\!28}{70\!\cdots\!71}$, $\frac{68\!\cdots\!63}{17\!\cdots\!76}a^{15}-\frac{36\!\cdots\!81}{17\!\cdots\!76}a^{14}+\frac{56\!\cdots\!57}{89\!\cdots\!88}a^{13}-\frac{28\!\cdots\!79}{17\!\cdots\!76}a^{12}+\frac{24\!\cdots\!05}{17\!\cdots\!76}a^{11}-\frac{11\!\cdots\!61}{17\!\cdots\!76}a^{10}+\frac{11\!\cdots\!35}{44\!\cdots\!44}a^{9}-\frac{15\!\cdots\!61}{17\!\cdots\!76}a^{8}+\frac{41\!\cdots\!63}{17\!\cdots\!76}a^{7}-\frac{29\!\cdots\!67}{44\!\cdots\!44}a^{6}+\frac{64\!\cdots\!47}{44\!\cdots\!44}a^{5}-\frac{15\!\cdots\!29}{56\!\cdots\!68}a^{4}+\frac{39\!\cdots\!17}{11\!\cdots\!36}a^{3}-\frac{11\!\cdots\!19}{18\!\cdots\!28}a^{2}+\frac{12\!\cdots\!33}{14\!\cdots\!42}a-\frac{16\!\cdots\!17}{70\!\cdots\!71}$, $\frac{37\!\cdots\!87}{35\!\cdots\!52}a^{15}-\frac{28\!\cdots\!65}{35\!\cdots\!52}a^{14}+\frac{49\!\cdots\!65}{17\!\cdots\!76}a^{13}-\frac{22\!\cdots\!11}{35\!\cdots\!52}a^{12}+\frac{14\!\cdots\!65}{35\!\cdots\!52}a^{11}-\frac{92\!\cdots\!05}{35\!\cdots\!52}a^{10}+\frac{94\!\cdots\!99}{89\!\cdots\!88}a^{9}-\frac{12\!\cdots\!25}{35\!\cdots\!52}a^{8}+\frac{31\!\cdots\!19}{35\!\cdots\!52}a^{7}-\frac{22\!\cdots\!43}{89\!\cdots\!88}a^{6}+\frac{50\!\cdots\!01}{89\!\cdots\!88}a^{5}-\frac{12\!\cdots\!71}{11\!\cdots\!36}a^{4}+\frac{30\!\cdots\!05}{22\!\cdots\!72}a^{3}-\frac{89\!\cdots\!67}{36\!\cdots\!56}a^{2}+\frac{10\!\cdots\!29}{28\!\cdots\!84}a-\frac{56\!\cdots\!43}{70\!\cdots\!71}$, $\frac{28\!\cdots\!65}{74\!\cdots\!64}a^{15}-\frac{55\!\cdots\!63}{74\!\cdots\!64}a^{14}+\frac{95\!\cdots\!75}{37\!\cdots\!32}a^{13}-\frac{50\!\cdots\!37}{74\!\cdots\!64}a^{12}+\frac{11\!\cdots\!91}{74\!\cdots\!64}a^{11}-\frac{18\!\cdots\!19}{74\!\cdots\!64}a^{10}+\frac{15\!\cdots\!11}{18\!\cdots\!16}a^{9}-\frac{25\!\cdots\!95}{74\!\cdots\!64}a^{8}+\frac{73\!\cdots\!21}{74\!\cdots\!64}a^{7}-\frac{52\!\cdots\!75}{18\!\cdots\!16}a^{6}+\frac{12\!\cdots\!37}{18\!\cdots\!16}a^{5}-\frac{71\!\cdots\!93}{46\!\cdots\!04}a^{4}+\frac{79\!\cdots\!91}{46\!\cdots\!04}a^{3}-\frac{60\!\cdots\!95}{23\!\cdots\!52}a^{2}+\frac{67\!\cdots\!98}{14\!\cdots\!47}a-\frac{26\!\cdots\!23}{39\!\cdots\!31}$, $\frac{40\!\cdots\!01}{74\!\cdots\!64}a^{15}+\frac{54\!\cdots\!19}{74\!\cdots\!64}a^{14}-\frac{11\!\cdots\!65}{46\!\cdots\!04}a^{13}+\frac{48\!\cdots\!71}{74\!\cdots\!64}a^{12}-\frac{93\!\cdots\!15}{74\!\cdots\!64}a^{11}+\frac{18\!\cdots\!91}{74\!\cdots\!64}a^{10}-\frac{27\!\cdots\!61}{37\!\cdots\!32}a^{9}+\frac{24\!\cdots\!05}{74\!\cdots\!64}a^{8}-\frac{68\!\cdots\!05}{74\!\cdots\!64}a^{7}+\frac{10\!\cdots\!87}{37\!\cdots\!32}a^{6}-\frac{58\!\cdots\!81}{92\!\cdots\!08}a^{5}+\frac{66\!\cdots\!49}{46\!\cdots\!04}a^{4}-\frac{73\!\cdots\!77}{46\!\cdots\!04}a^{3}+\frac{28\!\cdots\!03}{11\!\cdots\!76}a^{2}-\frac{24\!\cdots\!65}{58\!\cdots\!88}a+\frac{24\!\cdots\!97}{39\!\cdots\!31}$, $\frac{21\!\cdots\!31}{74\!\cdots\!64}a^{15}-\frac{41\!\cdots\!69}{74\!\cdots\!64}a^{14}+\frac{65\!\cdots\!71}{37\!\cdots\!32}a^{13}-\frac{38\!\cdots\!79}{74\!\cdots\!64}a^{12}+\frac{10\!\cdots\!29}{74\!\cdots\!64}a^{11}-\frac{12\!\cdots\!53}{74\!\cdots\!64}a^{10}+\frac{52\!\cdots\!87}{92\!\cdots\!08}a^{9}-\frac{18\!\cdots\!49}{74\!\cdots\!64}a^{8}+\frac{41\!\cdots\!15}{74\!\cdots\!64}a^{7}-\frac{64\!\cdots\!23}{46\!\cdots\!04}a^{6}+\frac{13\!\cdots\!33}{59\!\cdots\!36}a^{5}-\frac{22\!\cdots\!71}{11\!\cdots\!76}a^{4}-\frac{50\!\cdots\!73}{78\!\cdots\!56}a^{3}+\frac{67\!\cdots\!65}{23\!\cdots\!52}a^{2}-\frac{34\!\cdots\!59}{58\!\cdots\!88}a+\frac{20\!\cdots\!95}{39\!\cdots\!31}$, $\frac{29\!\cdots\!23}{74\!\cdots\!64}a^{15}-\frac{35\!\cdots\!97}{74\!\cdots\!64}a^{14}+\frac{87\!\cdots\!37}{37\!\cdots\!32}a^{13}-\frac{25\!\cdots\!97}{12\!\cdots\!96}a^{12}+\frac{94\!\cdots\!17}{74\!\cdots\!64}a^{11}-\frac{12\!\cdots\!57}{74\!\cdots\!64}a^{10}+\frac{26\!\cdots\!57}{18\!\cdots\!16}a^{9}-\frac{14\!\cdots\!53}{74\!\cdots\!64}a^{8}+\frac{67\!\cdots\!15}{74\!\cdots\!64}a^{7}-\frac{27\!\cdots\!85}{18\!\cdots\!16}a^{6}+\frac{26\!\cdots\!79}{59\!\cdots\!36}a^{5}-\frac{13\!\cdots\!49}{11\!\cdots\!76}a^{4}+\frac{60\!\cdots\!01}{46\!\cdots\!04}a^{3}+\frac{35\!\cdots\!75}{23\!\cdots\!52}a^{2}+\frac{15\!\cdots\!63}{58\!\cdots\!88}a+\frac{79\!\cdots\!42}{39\!\cdots\!31}$ (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: | \( 4516212.78969 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4516212.78969 \cdot 33}{2\cdot\sqrt{1077123334824966013513439398801}}\cr\approx \mathstrut & 0.174407264376 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2:C_8$ (as 16T24):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_2^2 : C_8$ |
Character table for $C_2^2 : C_8$ |
Intermediate fields
\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.0.5031233.1, 4.0.122713.1, 8.0.194754273881.1, 8.8.1037845525511849.1, 8.0.25313305500289.1 |
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: | 16.0.30588408049083077668563908786045909041.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(41\) | 41.16.14.1 | $x^{16} + 304 x^{15} + 40480 x^{14} + 3085600 x^{13} + 147416080 x^{12} + 4529584192 x^{11} + 87831092608 x^{10} + 996302227840 x^{9} + 5391168776882 x^{8} + 5977813379504 x^{7} + 3161920977824 x^{6} + 978514601120 x^{5} + 196936323920 x^{4} + 202153692608 x^{3} + 3372805705856 x^{2} + 36445904670848 x + 172395305267889$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |
\(73\) | 73.4.0.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
73.4.0.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
73.8.4.1 | $x^{8} + 15768 x^{7} + 93236508 x^{6} + 245028523288 x^{5} + 241487187554464 x^{4} + 21812691357056 x^{3} + 3877844534238648 x^{2} + 13666761747168624 x + 1459074653762756$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |