Normalized defining polynomial
\( x^{16} - 8 x^{15} + 32 x^{14} - 78 x^{13} + 130 x^{12} - 166 x^{11} + 268 x^{10} - 514 x^{9} + \cdots + 111 \)
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: | \(2555478887462114090409\) \(\medspace = 3^{12}\cdot 37^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.78\) | 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: | $3^{3/4}37^{3/4}\approx 34.197332740531884$ | ||
Ramified primes: | \(3\), \(37\) | 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 not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{12}-\frac{1}{6}a^{10}+\frac{1}{3}a^{6}-\frac{1}{2}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{42}a^{13}+\frac{1}{42}a^{12}+\frac{4}{21}a^{11}+\frac{1}{21}a^{10}+\frac{3}{14}a^{9}+\frac{3}{14}a^{8}-\frac{13}{42}a^{7}+\frac{4}{21}a^{6}-\frac{1}{3}a^{5}+\frac{1}{6}a^{4}+\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{2394}a^{14}+\frac{10}{1197}a^{13}-\frac{1}{42}a^{12}-\frac{17}{342}a^{11}+\frac{223}{1197}a^{10}-\frac{17}{798}a^{9}-\frac{299}{1197}a^{8}+\frac{311}{1197}a^{7}-\frac{185}{798}a^{6}-\frac{91}{342}a^{5}+\frac{169}{342}a^{4}-\frac{1}{42}a^{3}-\frac{134}{399}a^{2}+\frac{187}{798}a-\frac{16}{399}$, $\frac{1}{11\!\cdots\!78}a^{15}+\frac{11470158425615}{11\!\cdots\!78}a^{14}-\frac{208364600986285}{19\!\cdots\!63}a^{13}-\frac{18\!\cdots\!48}{58\!\cdots\!89}a^{12}+\frac{14\!\cdots\!65}{11\!\cdots\!78}a^{11}+\frac{604806680695531}{39\!\cdots\!26}a^{10}-\frac{88\!\cdots\!52}{58\!\cdots\!89}a^{9}-\frac{195982460543071}{30\!\cdots\!31}a^{8}+\frac{40\!\cdots\!02}{19\!\cdots\!63}a^{7}-\frac{67\!\cdots\!95}{11\!\cdots\!78}a^{6}+\frac{323101079608945}{884885177599866}a^{5}+\frac{18\!\cdots\!35}{19\!\cdots\!63}a^{4}-\frac{57\!\cdots\!77}{39\!\cdots\!26}a^{3}-\frac{39\!\cdots\!47}{39\!\cdots\!26}a^{2}+\frac{70\!\cdots\!23}{39\!\cdots\!26}a+\frac{13\!\cdots\!59}{65\!\cdots\!21}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{4}$, which has order $4$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{29307892456}{986651201529} a^{15} - \frac{82456597487}{328883733843} a^{14} + \frac{1048469024840}{986651201529} a^{13} - \frac{5520653047591}{1973302403058} a^{12} + \frac{3383118459211}{657767467686} a^{11} - \frac{14436526854047}{1973302403058} a^{10} + \frac{22484806508203}{1973302403058} a^{9} - \frac{6773284311190}{328883733843} a^{8} + \frac{52110837920273}{1973302403058} a^{7} - \frac{4033606006435}{140950171647} a^{6} + \frac{719033291315}{93966781098} a^{5} + \frac{27156090258826}{986651201529} a^{4} - \frac{6014111051743}{657767467686} a^{3} + \frac{3846996005777}{219255822562} a^{2} + \frac{9426403618505}{328883733843} a + \frac{785342047499}{93966781098} \) (order $6$) | 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: | $\frac{13\!\cdots\!25}{11\!\cdots\!78}a^{15}-\frac{19\!\cdots\!14}{19\!\cdots\!63}a^{14}+\frac{24\!\cdots\!67}{58\!\cdots\!89}a^{13}-\frac{12\!\cdots\!25}{11\!\cdots\!78}a^{12}+\frac{76\!\cdots\!91}{39\!\cdots\!26}a^{11}-\frac{16\!\cdots\!27}{58\!\cdots\!89}a^{10}+\frac{50\!\cdots\!85}{11\!\cdots\!78}a^{9}-\frac{15\!\cdots\!05}{19\!\cdots\!63}a^{8}+\frac{16\!\cdots\!95}{16\!\cdots\!54}a^{7}-\frac{12\!\cdots\!93}{11\!\cdots\!78}a^{6}+\frac{13\!\cdots\!95}{56\!\cdots\!18}a^{5}+\frac{12\!\cdots\!69}{11\!\cdots\!78}a^{4}-\frac{52\!\cdots\!99}{19\!\cdots\!63}a^{3}+\frac{38\!\cdots\!84}{65\!\cdots\!21}a^{2}+\frac{27\!\cdots\!11}{28\!\cdots\!09}a+\frac{61\!\cdots\!89}{19\!\cdots\!63}$, $\frac{250343319763}{144051075423234}a^{15}-\frac{6310010877039}{304107825893494}a^{14}+\frac{160385749811506}{13\!\cdots\!23}a^{13}-\frac{29937909295030}{72025537711617}a^{12}+\frac{44925000725177}{43443975127642}a^{11}-\frac{26\!\cdots\!45}{13\!\cdots\!23}a^{10}+\frac{43\!\cdots\!42}{13\!\cdots\!23}a^{9}-\frac{783424828063434}{152053912946747}a^{8}+\frac{16\!\cdots\!70}{195497888074389}a^{7}-\frac{15\!\cdots\!20}{13\!\cdots\!23}a^{6}+\frac{544639360467899}{43443975127642}a^{5}-\frac{21\!\cdots\!51}{27\!\cdots\!46}a^{4}-\frac{4947671726065}{24008512570539}a^{3}+\frac{462423943435404}{152053912946747}a^{2}-\frac{479513783760943}{130331925382926}a-\frac{613736188688524}{456161738840241}$, $\frac{11\!\cdots\!31}{11\!\cdots\!78}a^{15}-\frac{554795705005669}{65\!\cdots\!21}a^{14}+\frac{42\!\cdots\!19}{11\!\cdots\!78}a^{13}-\frac{57\!\cdots\!99}{58\!\cdots\!89}a^{12}+\frac{36\!\cdots\!15}{19\!\cdots\!63}a^{11}-\frac{82\!\cdots\!88}{30\!\cdots\!31}a^{10}+\frac{48\!\cdots\!71}{11\!\cdots\!78}a^{9}-\frac{13\!\cdots\!33}{18\!\cdots\!06}a^{8}+\frac{11\!\cdots\!55}{11\!\cdots\!78}a^{7}-\frac{12\!\cdots\!37}{11\!\cdots\!78}a^{6}+\frac{23\!\cdots\!13}{56\!\cdots\!18}a^{5}+\frac{98\!\cdots\!93}{11\!\cdots\!78}a^{4}-\frac{74\!\cdots\!23}{19\!\cdots\!63}a^{3}+\frac{92\!\cdots\!93}{13\!\cdots\!42}a^{2}+\frac{32\!\cdots\!01}{39\!\cdots\!26}a+\frac{34\!\cdots\!79}{19\!\cdots\!63}$, $\frac{610420931890583}{11\!\cdots\!78}a^{15}-\frac{23\!\cdots\!79}{58\!\cdots\!89}a^{14}+\frac{18\!\cdots\!97}{11\!\cdots\!78}a^{13}-\frac{39\!\cdots\!45}{11\!\cdots\!78}a^{12}+\frac{52\!\cdots\!49}{11\!\cdots\!78}a^{11}-\frac{35\!\cdots\!19}{11\!\cdots\!78}a^{10}+\frac{23\!\cdots\!10}{58\!\cdots\!89}a^{9}-\frac{13\!\cdots\!03}{11\!\cdots\!78}a^{8}+\frac{24\!\cdots\!77}{58\!\cdots\!89}a^{7}+\frac{16\!\cdots\!73}{11\!\cdots\!78}a^{6}-\frac{55\!\cdots\!96}{84\!\cdots\!27}a^{5}+\frac{12\!\cdots\!81}{11\!\cdots\!78}a^{4}-\frac{22\!\cdots\!83}{13\!\cdots\!42}a^{3}+\frac{23\!\cdots\!66}{19\!\cdots\!63}a^{2}+\frac{10\!\cdots\!45}{13\!\cdots\!42}a+\frac{82\!\cdots\!91}{39\!\cdots\!26}$, $\frac{40\!\cdots\!10}{58\!\cdots\!89}a^{15}-\frac{67\!\cdots\!15}{11\!\cdots\!78}a^{14}+\frac{47\!\cdots\!99}{19\!\cdots\!63}a^{13}-\frac{75\!\cdots\!41}{11\!\cdots\!78}a^{12}+\frac{68\!\cdots\!35}{58\!\cdots\!89}a^{11}-\frac{31\!\cdots\!38}{19\!\cdots\!63}a^{10}+\frac{29\!\cdots\!09}{11\!\cdots\!78}a^{9}-\frac{53\!\cdots\!33}{11\!\cdots\!78}a^{8}+\frac{22\!\cdots\!27}{39\!\cdots\!26}a^{7}-\frac{72\!\cdots\!69}{11\!\cdots\!78}a^{6}+\frac{96\!\cdots\!99}{84\!\cdots\!27}a^{5}+\frac{27\!\cdots\!59}{39\!\cdots\!26}a^{4}-\frac{49\!\cdots\!12}{19\!\cdots\!63}a^{3}+\frac{16\!\cdots\!59}{39\!\cdots\!26}a^{2}+\frac{13\!\cdots\!83}{19\!\cdots\!63}a+\frac{21\!\cdots\!27}{13\!\cdots\!42}$, $\frac{23\!\cdots\!23}{11\!\cdots\!78}a^{15}-\frac{19\!\cdots\!63}{11\!\cdots\!78}a^{14}+\frac{26\!\cdots\!53}{39\!\cdots\!26}a^{13}-\frac{14\!\cdots\!21}{84\!\cdots\!27}a^{12}+\frac{37\!\cdots\!29}{11\!\cdots\!78}a^{11}-\frac{89\!\cdots\!23}{20\!\cdots\!54}a^{10}+\frac{39\!\cdots\!43}{58\!\cdots\!89}a^{9}-\frac{21\!\cdots\!29}{16\!\cdots\!54}a^{8}+\frac{60\!\cdots\!23}{39\!\cdots\!26}a^{7}-\frac{94\!\cdots\!32}{58\!\cdots\!89}a^{6}+\frac{34\!\cdots\!87}{16\!\cdots\!54}a^{5}+\frac{79\!\cdots\!49}{39\!\cdots\!26}a^{4}-\frac{95\!\cdots\!42}{19\!\cdots\!63}a^{3}+\frac{40\!\cdots\!23}{39\!\cdots\!26}a^{2}+\frac{82\!\cdots\!45}{39\!\cdots\!26}a+\frac{78\!\cdots\!69}{13\!\cdots\!42}$, $\frac{11\!\cdots\!78}{58\!\cdots\!89}a^{15}-\frac{34\!\cdots\!45}{19\!\cdots\!63}a^{14}+\frac{48\!\cdots\!75}{61\!\cdots\!62}a^{13}-\frac{26\!\cdots\!41}{11\!\cdots\!78}a^{12}+\frac{85\!\cdots\!53}{18\!\cdots\!06}a^{11}-\frac{42\!\cdots\!11}{58\!\cdots\!89}a^{10}+\frac{67\!\cdots\!71}{58\!\cdots\!89}a^{9}-\frac{38\!\cdots\!72}{19\!\cdots\!63}a^{8}+\frac{16\!\cdots\!37}{58\!\cdots\!89}a^{7}-\frac{20\!\cdots\!09}{58\!\cdots\!89}a^{6}+\frac{22\!\cdots\!29}{934045465244303}a^{5}+\frac{24\!\cdots\!67}{61\!\cdots\!62}a^{4}-\frac{10\!\cdots\!58}{19\!\cdots\!63}a^{3}+\frac{11\!\cdots\!54}{65\!\cdots\!21}a^{2}+\frac{22\!\cdots\!89}{19\!\cdots\!63}a+\frac{37\!\cdots\!77}{20\!\cdots\!54}$ | 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: | \( 34293.5127133 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 34293.5127133 \cdot 4}{6\cdot\sqrt{2555478887462114090409}}\cr\approx \mathstrut & 1.09855931691 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 16T33):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3:C_4$ |
Character table for $C_2^3:C_4$ |
Intermediate fields
\(\Q(\sqrt{-111}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{37}) \), 4.2.4107.1 x2, 4.0.333.1 x2, \(\Q(\sqrt{-3}, \sqrt{37})\), 8.0.151807041.1, 8.0.1366263369.1 x2, 8.0.50551744653.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(37\) | 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.4.3.2 | $x^{4} + 37$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
37.4.3.2 | $x^{4} + 37$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |