Normalized defining polynomial
\( x^{16} - x^{15} - 22 x^{14} - 104 x^{13} - 55 x^{12} + 6168 x^{11} - 19527 x^{10} - 54968 x^{9} + \cdots - 364019 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(95581759383007560835666012898929\) \(\medspace = 11^{4}\cdot 97^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(99.72\) | 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: | $11^{1/2}97^{7/8}\approx 181.60241101432212$ | ||
Ramified primes: | \(11\), \(97\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{83\!\cdots\!06}a^{15}-\frac{19\!\cdots\!79}{83\!\cdots\!06}a^{14}+\frac{86\!\cdots\!52}{41\!\cdots\!53}a^{13}-\frac{85\!\cdots\!02}{41\!\cdots\!53}a^{12}-\frac{86\!\cdots\!86}{41\!\cdots\!53}a^{11}+\frac{94\!\cdots\!70}{41\!\cdots\!53}a^{10}-\frac{40\!\cdots\!03}{83\!\cdots\!06}a^{9}-\frac{10\!\cdots\!55}{41\!\cdots\!53}a^{8}-\frac{30\!\cdots\!87}{83\!\cdots\!06}a^{7}-\frac{11\!\cdots\!48}{41\!\cdots\!53}a^{6}-\frac{18\!\cdots\!11}{41\!\cdots\!53}a^{5}+\frac{16\!\cdots\!09}{83\!\cdots\!06}a^{4}-\frac{33\!\cdots\!15}{83\!\cdots\!06}a^{3}+\frac{45\!\cdots\!64}{41\!\cdots\!53}a^{2}-\frac{31\!\cdots\!31}{83\!\cdots\!06}a-\frac{12\!\cdots\!95}{41\!\cdots\!53}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{70\!\cdots\!25}{23\!\cdots\!39}a^{15}+\frac{57\!\cdots\!49}{47\!\cdots\!78}a^{14}-\frac{29\!\cdots\!15}{47\!\cdots\!78}a^{13}-\frac{18\!\cdots\!41}{47\!\cdots\!78}a^{12}-\frac{16\!\cdots\!43}{23\!\cdots\!39}a^{11}+\frac{81\!\cdots\!39}{47\!\cdots\!78}a^{10}-\frac{83\!\cdots\!55}{23\!\cdots\!39}a^{9}-\frac{10\!\cdots\!49}{47\!\cdots\!78}a^{8}+\frac{58\!\cdots\!19}{47\!\cdots\!78}a^{7}-\frac{25\!\cdots\!79}{23\!\cdots\!39}a^{6}-\frac{15\!\cdots\!25}{23\!\cdots\!39}a^{5}+\frac{85\!\cdots\!31}{47\!\cdots\!78}a^{4}-\frac{27\!\cdots\!41}{23\!\cdots\!39}a^{3}-\frac{24\!\cdots\!85}{23\!\cdots\!39}a^{2}+\frac{13\!\cdots\!52}{23\!\cdots\!39}a+\frac{41\!\cdots\!61}{47\!\cdots\!78}$, $\frac{52\!\cdots\!71}{21\!\cdots\!21}a^{15}-\frac{84\!\cdots\!57}{43\!\cdots\!42}a^{14}-\frac{11\!\cdots\!84}{21\!\cdots\!21}a^{13}-\frac{57\!\cdots\!99}{21\!\cdots\!21}a^{12}-\frac{76\!\cdots\!15}{43\!\cdots\!42}a^{11}+\frac{64\!\cdots\!91}{43\!\cdots\!42}a^{10}-\frac{95\!\cdots\!77}{21\!\cdots\!21}a^{9}-\frac{62\!\cdots\!03}{43\!\cdots\!42}a^{8}+\frac{52\!\cdots\!79}{43\!\cdots\!42}a^{7}-\frac{86\!\cdots\!91}{43\!\cdots\!42}a^{6}-\frac{97\!\cdots\!30}{21\!\cdots\!21}a^{5}+\frac{45\!\cdots\!48}{21\!\cdots\!21}a^{4}-\frac{53\!\cdots\!55}{21\!\cdots\!21}a^{3}+\frac{14\!\cdots\!14}{21\!\cdots\!21}a^{2}+\frac{50\!\cdots\!91}{43\!\cdots\!42}a+\frac{29\!\cdots\!98}{21\!\cdots\!21}$, $\frac{95\!\cdots\!22}{21\!\cdots\!21}a^{15}-\frac{38\!\cdots\!09}{21\!\cdots\!21}a^{14}-\frac{27\!\cdots\!27}{21\!\cdots\!21}a^{13}-\frac{51\!\cdots\!36}{21\!\cdots\!21}a^{12}+\frac{40\!\cdots\!42}{21\!\cdots\!21}a^{11}+\frac{14\!\cdots\!17}{43\!\cdots\!42}a^{10}-\frac{32\!\cdots\!73}{21\!\cdots\!21}a^{9}-\frac{81\!\cdots\!05}{43\!\cdots\!42}a^{8}+\frac{69\!\cdots\!10}{21\!\cdots\!21}a^{7}-\frac{17\!\cdots\!81}{21\!\cdots\!21}a^{6}-\frac{33\!\cdots\!67}{43\!\cdots\!42}a^{5}+\frac{27\!\cdots\!23}{43\!\cdots\!42}a^{4}-\frac{20\!\cdots\!00}{21\!\cdots\!21}a^{3}+\frac{73\!\cdots\!47}{43\!\cdots\!42}a^{2}+\frac{18\!\cdots\!67}{43\!\cdots\!42}a+\frac{90\!\cdots\!31}{43\!\cdots\!42}$, $\frac{18\!\cdots\!17}{21\!\cdots\!21}a^{15}+\frac{34\!\cdots\!81}{21\!\cdots\!21}a^{14}-\frac{71\!\cdots\!23}{43\!\cdots\!42}a^{13}-\frac{60\!\cdots\!63}{43\!\cdots\!42}a^{12}-\frac{87\!\cdots\!80}{21\!\cdots\!21}a^{11}+\frac{19\!\cdots\!61}{43\!\cdots\!42}a^{10}-\frac{91\!\cdots\!09}{43\!\cdots\!42}a^{9}-\frac{28\!\cdots\!21}{43\!\cdots\!42}a^{8}+\frac{51\!\cdots\!82}{21\!\cdots\!21}a^{7}+\frac{24\!\cdots\!85}{21\!\cdots\!21}a^{6}-\frac{37\!\cdots\!23}{21\!\cdots\!21}a^{5}+\frac{44\!\cdots\!42}{21\!\cdots\!21}a^{4}+\frac{25\!\cdots\!25}{43\!\cdots\!42}a^{3}-\frac{25\!\cdots\!14}{21\!\cdots\!21}a^{2}+\frac{30\!\cdots\!71}{43\!\cdots\!42}a+\frac{20\!\cdots\!47}{21\!\cdots\!21}$, $\frac{13\!\cdots\!40}{21\!\cdots\!21}a^{15}+\frac{40\!\cdots\!65}{43\!\cdots\!42}a^{14}-\frac{54\!\cdots\!29}{43\!\cdots\!42}a^{13}-\frac{21\!\cdots\!99}{21\!\cdots\!21}a^{12}-\frac{11\!\cdots\!19}{43\!\cdots\!42}a^{11}+\frac{14\!\cdots\!97}{43\!\cdots\!42}a^{10}-\frac{89\!\cdots\!05}{43\!\cdots\!42}a^{9}-\frac{98\!\cdots\!30}{21\!\cdots\!21}a^{8}+\frac{40\!\cdots\!22}{21\!\cdots\!21}a^{7}+\frac{43\!\cdots\!61}{21\!\cdots\!21}a^{6}-\frac{25\!\cdots\!08}{21\!\cdots\!21}a^{5}+\frac{70\!\cdots\!59}{43\!\cdots\!42}a^{4}-\frac{17\!\cdots\!38}{21\!\cdots\!21}a^{3}+\frac{39\!\cdots\!13}{43\!\cdots\!42}a^{2}+\frac{14\!\cdots\!99}{21\!\cdots\!21}a-\frac{70\!\cdots\!16}{21\!\cdots\!21}$, $\frac{12\!\cdots\!35}{43\!\cdots\!42}a^{15}-\frac{11\!\cdots\!06}{21\!\cdots\!21}a^{14}-\frac{26\!\cdots\!15}{43\!\cdots\!42}a^{13}-\frac{15\!\cdots\!21}{43\!\cdots\!42}a^{12}-\frac{10\!\cdots\!76}{21\!\cdots\!21}a^{11}+\frac{36\!\cdots\!36}{21\!\cdots\!21}a^{10}-\frac{18\!\cdots\!59}{43\!\cdots\!42}a^{9}-\frac{37\!\cdots\!27}{21\!\cdots\!21}a^{8}+\frac{27\!\cdots\!59}{21\!\cdots\!21}a^{7}-\frac{39\!\cdots\!65}{21\!\cdots\!21}a^{6}-\frac{21\!\cdots\!89}{43\!\cdots\!42}a^{5}+\frac{88\!\cdots\!25}{43\!\cdots\!42}a^{4}-\frac{10\!\cdots\!45}{43\!\cdots\!42}a^{3}+\frac{19\!\cdots\!29}{43\!\cdots\!42}a^{2}+\frac{26\!\cdots\!42}{21\!\cdots\!21}a-\frac{92\!\cdots\!54}{21\!\cdots\!21}$, $\frac{30\!\cdots\!99}{83\!\cdots\!06}a^{15}+\frac{14\!\cdots\!91}{41\!\cdots\!53}a^{14}-\frac{60\!\cdots\!89}{83\!\cdots\!06}a^{13}-\frac{21\!\cdots\!63}{41\!\cdots\!53}a^{12}-\frac{10\!\cdots\!45}{83\!\cdots\!06}a^{11}+\frac{83\!\cdots\!05}{41\!\cdots\!53}a^{10}-\frac{14\!\cdots\!39}{41\!\cdots\!53}a^{9}-\frac{21\!\cdots\!29}{83\!\cdots\!06}a^{8}+\frac{11\!\cdots\!31}{83\!\cdots\!06}a^{7}-\frac{39\!\cdots\!07}{41\!\cdots\!53}a^{6}-\frac{61\!\cdots\!99}{83\!\cdots\!06}a^{5}+\frac{15\!\cdots\!21}{83\!\cdots\!06}a^{4}-\frac{10\!\cdots\!25}{83\!\cdots\!06}a^{3}-\frac{29\!\cdots\!21}{41\!\cdots\!53}a^{2}+\frac{18\!\cdots\!56}{41\!\cdots\!53}a+\frac{42\!\cdots\!07}{83\!\cdots\!06}$, $\frac{15\!\cdots\!81}{21\!\cdots\!21}a^{15}+\frac{22\!\cdots\!07}{21\!\cdots\!21}a^{14}-\frac{31\!\cdots\!30}{21\!\cdots\!21}a^{13}-\frac{49\!\cdots\!47}{43\!\cdots\!42}a^{12}-\frac{12\!\cdots\!49}{43\!\cdots\!42}a^{11}+\frac{86\!\cdots\!80}{21\!\cdots\!21}a^{10}-\frac{16\!\cdots\!71}{43\!\cdots\!42}a^{9}-\frac{23\!\cdots\!25}{43\!\cdots\!42}a^{8}+\frac{10\!\cdots\!07}{43\!\cdots\!42}a^{7}-\frac{23\!\cdots\!20}{21\!\cdots\!21}a^{6}-\frac{32\!\cdots\!25}{21\!\cdots\!21}a^{5}+\frac{54\!\cdots\!93}{21\!\cdots\!21}a^{4}-\frac{11\!\cdots\!93}{21\!\cdots\!21}a^{3}-\frac{55\!\cdots\!93}{43\!\cdots\!42}a^{2}+\frac{10\!\cdots\!35}{21\!\cdots\!21}a+\frac{25\!\cdots\!13}{43\!\cdots\!42}$, $\frac{17\!\cdots\!62}{41\!\cdots\!53}a^{15}-\frac{42\!\cdots\!65}{41\!\cdots\!53}a^{14}-\frac{32\!\cdots\!03}{41\!\cdots\!53}a^{13}-\frac{12\!\cdots\!92}{41\!\cdots\!53}a^{12}+\frac{13\!\cdots\!27}{41\!\cdots\!53}a^{11}+\frac{20\!\cdots\!25}{83\!\cdots\!06}a^{10}-\frac{49\!\cdots\!40}{41\!\cdots\!53}a^{9}-\frac{62\!\cdots\!51}{83\!\cdots\!06}a^{8}+\frac{96\!\cdots\!70}{41\!\cdots\!53}a^{7}-\frac{29\!\cdots\!46}{41\!\cdots\!53}a^{6}+\frac{12\!\cdots\!27}{83\!\cdots\!06}a^{5}+\frac{34\!\cdots\!69}{83\!\cdots\!06}a^{4}-\frac{45\!\cdots\!69}{41\!\cdots\!53}a^{3}+\frac{10\!\cdots\!89}{83\!\cdots\!06}a^{2}-\frac{60\!\cdots\!87}{83\!\cdots\!06}a+\frac{12\!\cdots\!47}{83\!\cdots\!06}$, $\frac{19\!\cdots\!05}{83\!\cdots\!06}a^{15}+\frac{58\!\cdots\!09}{83\!\cdots\!06}a^{14}-\frac{43\!\cdots\!51}{83\!\cdots\!06}a^{13}-\frac{13\!\cdots\!27}{41\!\cdots\!53}a^{12}-\frac{22\!\cdots\!37}{41\!\cdots\!53}a^{11}+\frac{59\!\cdots\!44}{41\!\cdots\!53}a^{10}-\frac{11\!\cdots\!91}{41\!\cdots\!53}a^{9}-\frac{14\!\cdots\!33}{83\!\cdots\!06}a^{8}+\frac{41\!\cdots\!21}{41\!\cdots\!53}a^{7}-\frac{73\!\cdots\!17}{83\!\cdots\!06}a^{6}-\frac{21\!\cdots\!59}{41\!\cdots\!53}a^{5}+\frac{59\!\cdots\!40}{41\!\cdots\!53}a^{4}-\frac{78\!\cdots\!23}{83\!\cdots\!06}a^{3}-\frac{52\!\cdots\!63}{83\!\cdots\!06}a^{2}+\frac{29\!\cdots\!84}{41\!\cdots\!53}a+\frac{25\!\cdots\!03}{41\!\cdots\!53}$, $\frac{54\!\cdots\!77}{83\!\cdots\!06}a^{15}-\frac{69\!\cdots\!57}{41\!\cdots\!53}a^{14}-\frac{84\!\cdots\!21}{83\!\cdots\!06}a^{13}-\frac{44\!\cdots\!63}{83\!\cdots\!06}a^{12}-\frac{21\!\cdots\!81}{83\!\cdots\!06}a^{11}+\frac{16\!\cdots\!18}{41\!\cdots\!53}a^{10}-\frac{81\!\cdots\!88}{41\!\cdots\!53}a^{9}+\frac{27\!\cdots\!89}{41\!\cdots\!53}a^{8}+\frac{11\!\cdots\!96}{41\!\cdots\!53}a^{7}-\frac{97\!\cdots\!27}{83\!\cdots\!06}a^{6}+\frac{82\!\cdots\!18}{41\!\cdots\!53}a^{5}-\frac{77\!\cdots\!11}{83\!\cdots\!06}a^{4}-\frac{60\!\cdots\!68}{41\!\cdots\!53}a^{3}+\frac{77\!\cdots\!97}{41\!\cdots\!53}a^{2}-\frac{37\!\cdots\!35}{83\!\cdots\!06}a-\frac{30\!\cdots\!68}{41\!\cdots\!53}$ (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: | \( 8828543184.49 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 8828543184.49 \cdot 1}{2\cdot\sqrt{95581759383007560835666012898929}}\cr\approx \mathstrut & 0.180148636163 \end{aligned}\] (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
\(\Q(\sqrt{97}) \), 4.2.10039403.1, 4.4.912673.1, 4.2.103499.1, 8.4.9776592421851673.1, 8.8.80798284478113.1, 8.4.100789612596409.1 |
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: | 16.0.1399412539126613698194986094853219489.2 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(97\) | 97.16.14.1 | $x^{16} + 768 x^{15} + 258088 x^{14} + 49572096 x^{13} + 5953168060 x^{12} + 457847744256 x^{11} + 22036223404888 x^{10} + 608019502810368 x^{9} + 7433960430005160 x^{8} + 3040097514126336 x^{7} + 550905610125696 x^{6} + 57235766095872 x^{5} + 4296185158440 x^{4} + 44325875768832 x^{3} + 2119059344887056 x^{2} + 58091109365638656 x + 696714908933731396$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |