Normalized defining polynomial
\( x^{16} - 3 x^{15} + 11 x^{14} - 11 x^{13} + 59 x^{12} + 28 x^{11} + 249 x^{10} + 217 x^{9} + 546 x^{8} + \cdots + 16 \)
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: | \(4809039104363049933169\) \(\medspace = 13^{4}\cdot 17^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.65\) | 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: | $13^{1/2}17^{7/8}\approx 43.014460711706995$ | ||
Ramified primes: | \(13\), \(17\) | 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)
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This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{26\!\cdots\!68}a^{15}-\frac{10\!\cdots\!19}{26\!\cdots\!68}a^{14}+\frac{59\!\cdots\!83}{26\!\cdots\!68}a^{13}+\frac{30\!\cdots\!13}{26\!\cdots\!68}a^{12}-\frac{27\!\cdots\!81}{26\!\cdots\!68}a^{11}+\frac{14\!\cdots\!05}{66\!\cdots\!42}a^{10}+\frac{28\!\cdots\!65}{26\!\cdots\!68}a^{9}+\frac{11\!\cdots\!89}{26\!\cdots\!68}a^{8}+\frac{60\!\cdots\!33}{13\!\cdots\!84}a^{7}-\frac{28\!\cdots\!89}{13\!\cdots\!84}a^{6}+\frac{12\!\cdots\!09}{13\!\cdots\!84}a^{5}-\frac{11\!\cdots\!19}{26\!\cdots\!68}a^{4}+\frac{45\!\cdots\!33}{26\!\cdots\!68}a^{3}-\frac{37\!\cdots\!59}{33\!\cdots\!21}a^{2}+\frac{48\!\cdots\!63}{66\!\cdots\!42}a+\frac{12\!\cdots\!17}{33\!\cdots\!21}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( -1 \) (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: | $\frac{11\!\cdots\!51}{26\!\cdots\!68}a^{15}-\frac{27\!\cdots\!43}{26\!\cdots\!68}a^{14}+\frac{93\!\cdots\!35}{26\!\cdots\!68}a^{13}-\frac{68\!\cdots\!11}{26\!\cdots\!68}a^{12}+\frac{49\!\cdots\!79}{26\!\cdots\!68}a^{11}+\frac{44\!\cdots\!69}{13\!\cdots\!84}a^{10}+\frac{26\!\cdots\!63}{26\!\cdots\!68}a^{9}+\frac{41\!\cdots\!21}{26\!\cdots\!68}a^{8}+\frac{14\!\cdots\!73}{66\!\cdots\!42}a^{7}+\frac{18\!\cdots\!03}{13\!\cdots\!84}a^{6}+\frac{97\!\cdots\!87}{13\!\cdots\!84}a^{5}+\frac{39\!\cdots\!19}{26\!\cdots\!68}a^{4}+\frac{32\!\cdots\!45}{26\!\cdots\!68}a^{3}+\frac{18\!\cdots\!03}{13\!\cdots\!84}a^{2}+\frac{37\!\cdots\!15}{66\!\cdots\!42}a+\frac{57\!\cdots\!83}{33\!\cdots\!21}$, $\frac{41\!\cdots\!93}{26\!\cdots\!68}a^{15}-\frac{87\!\cdots\!73}{26\!\cdots\!68}a^{14}+\frac{30\!\cdots\!61}{26\!\cdots\!68}a^{13}+\frac{82\!\cdots\!47}{26\!\cdots\!68}a^{12}+\frac{16\!\cdots\!01}{26\!\cdots\!68}a^{11}+\frac{18\!\cdots\!67}{13\!\cdots\!84}a^{10}+\frac{93\!\cdots\!85}{26\!\cdots\!68}a^{9}+\frac{16\!\cdots\!35}{26\!\cdots\!68}a^{8}+\frac{54\!\cdots\!77}{66\!\cdots\!42}a^{7}+\frac{82\!\cdots\!39}{13\!\cdots\!84}a^{6}+\frac{35\!\cdots\!89}{13\!\cdots\!84}a^{5}+\frac{18\!\cdots\!93}{26\!\cdots\!68}a^{4}+\frac{13\!\cdots\!55}{26\!\cdots\!68}a^{3}+\frac{69\!\cdots\!15}{13\!\cdots\!84}a^{2}+\frac{69\!\cdots\!18}{33\!\cdots\!21}a-\frac{49\!\cdots\!50}{33\!\cdots\!21}$, $\frac{14\!\cdots\!09}{26\!\cdots\!68}a^{15}-\frac{35\!\cdots\!81}{26\!\cdots\!68}a^{14}+\frac{12\!\cdots\!85}{26\!\cdots\!68}a^{13}-\frac{50\!\cdots\!53}{26\!\cdots\!68}a^{12}+\frac{67\!\cdots\!77}{26\!\cdots\!68}a^{11}+\frac{46\!\cdots\!89}{13\!\cdots\!84}a^{10}+\frac{33\!\cdots\!01}{26\!\cdots\!68}a^{9}+\frac{46\!\cdots\!07}{26\!\cdots\!68}a^{8}+\frac{18\!\cdots\!75}{66\!\cdots\!42}a^{7}+\frac{21\!\cdots\!83}{13\!\cdots\!84}a^{6}+\frac{19\!\cdots\!31}{13\!\cdots\!84}a^{5}+\frac{44\!\cdots\!61}{26\!\cdots\!68}a^{4}+\frac{41\!\cdots\!03}{26\!\cdots\!68}a^{3}+\frac{24\!\cdots\!31}{13\!\cdots\!84}a^{2}+\frac{25\!\cdots\!43}{33\!\cdots\!21}a+\frac{90\!\cdots\!78}{33\!\cdots\!21}$, $\frac{12\!\cdots\!13}{66\!\cdots\!42}a^{15}-\frac{52\!\cdots\!61}{13\!\cdots\!84}a^{14}+\frac{17\!\cdots\!33}{13\!\cdots\!84}a^{13}+\frac{44\!\cdots\!45}{13\!\cdots\!84}a^{12}+\frac{91\!\cdots\!81}{13\!\cdots\!84}a^{11}+\frac{20\!\cdots\!81}{13\!\cdots\!84}a^{10}+\frac{12\!\cdots\!36}{33\!\cdots\!21}a^{9}+\frac{86\!\cdots\!87}{13\!\cdots\!84}a^{8}+\frac{10\!\cdots\!27}{13\!\cdots\!84}a^{7}+\frac{15\!\cdots\!19}{33\!\cdots\!21}a^{6}-\frac{14\!\cdots\!93}{33\!\cdots\!21}a^{5}+\frac{33\!\cdots\!49}{66\!\cdots\!42}a^{4}+\frac{48\!\cdots\!25}{13\!\cdots\!84}a^{3}+\frac{60\!\cdots\!43}{13\!\cdots\!84}a^{2}+\frac{62\!\cdots\!96}{33\!\cdots\!21}a-\frac{33\!\cdots\!45}{33\!\cdots\!21}$, $\frac{24\!\cdots\!66}{33\!\cdots\!21}a^{15}-\frac{24\!\cdots\!47}{13\!\cdots\!84}a^{14}+\frac{87\!\cdots\!59}{13\!\cdots\!84}a^{13}-\frac{41\!\cdots\!61}{13\!\cdots\!84}a^{12}+\frac{45\!\cdots\!13}{13\!\cdots\!84}a^{11}+\frac{57\!\cdots\!19}{13\!\cdots\!84}a^{10}+\frac{11\!\cdots\!69}{66\!\cdots\!42}a^{9}+\frac{29\!\cdots\!51}{13\!\cdots\!84}a^{8}+\frac{48\!\cdots\!59}{13\!\cdots\!84}a^{7}+\frac{63\!\cdots\!11}{33\!\cdots\!21}a^{6}+\frac{11\!\cdots\!19}{66\!\cdots\!42}a^{5}+\frac{61\!\cdots\!01}{33\!\cdots\!21}a^{4}+\frac{25\!\cdots\!01}{13\!\cdots\!84}a^{3}+\frac{30\!\cdots\!23}{13\!\cdots\!84}a^{2}+\frac{65\!\cdots\!75}{66\!\cdots\!42}a+\frac{15\!\cdots\!13}{33\!\cdots\!21}$, $\frac{64\!\cdots\!39}{13\!\cdots\!84}a^{15}-\frac{81\!\cdots\!13}{66\!\cdots\!42}a^{14}+\frac{14\!\cdots\!29}{33\!\cdots\!21}a^{13}-\frac{75\!\cdots\!99}{33\!\cdots\!21}a^{12}+\frac{78\!\cdots\!46}{33\!\cdots\!21}a^{11}+\frac{37\!\cdots\!91}{13\!\cdots\!84}a^{10}+\frac{15\!\cdots\!85}{13\!\cdots\!84}a^{9}+\frac{50\!\cdots\!43}{33\!\cdots\!21}a^{8}+\frac{34\!\cdots\!57}{13\!\cdots\!84}a^{7}+\frac{99\!\cdots\!89}{66\!\cdots\!42}a^{6}+\frac{52\!\cdots\!98}{33\!\cdots\!21}a^{5}+\frac{19\!\cdots\!17}{13\!\cdots\!84}a^{4}+\frac{97\!\cdots\!35}{66\!\cdots\!42}a^{3}+\frac{22\!\cdots\!57}{13\!\cdots\!84}a^{2}+\frac{23\!\cdots\!18}{33\!\cdots\!21}a+\frac{14\!\cdots\!27}{33\!\cdots\!21}$, $\frac{92\!\cdots\!81}{13\!\cdots\!84}a^{15}-\frac{22\!\cdots\!39}{13\!\cdots\!84}a^{14}+\frac{79\!\cdots\!11}{13\!\cdots\!84}a^{13}-\frac{25\!\cdots\!77}{13\!\cdots\!84}a^{12}+\frac{42\!\cdots\!59}{13\!\cdots\!84}a^{11}+\frac{15\!\cdots\!21}{33\!\cdots\!21}a^{10}+\frac{21\!\cdots\!53}{13\!\cdots\!84}a^{9}+\frac{30\!\cdots\!65}{13\!\cdots\!84}a^{8}+\frac{23\!\cdots\!35}{66\!\cdots\!42}a^{7}+\frac{14\!\cdots\!03}{66\!\cdots\!42}a^{6}+\frac{57\!\cdots\!12}{33\!\cdots\!21}a^{5}+\frac{29\!\cdots\!43}{13\!\cdots\!84}a^{4}+\frac{26\!\cdots\!27}{13\!\cdots\!84}a^{3}+\frac{76\!\cdots\!00}{33\!\cdots\!21}a^{2}+\frac{63\!\cdots\!99}{66\!\cdots\!42}a+\frac{95\!\cdots\!73}{33\!\cdots\!21}$ | 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: | \( 3640.01221338 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 4}{2\cdot\sqrt{4809039104363049933169}}\cr\approx \mathstrut & 0.255001437768 \end{aligned}\]
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
\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.0.3757.1, 4.0.63869.1, 8.0.69347235737.1, \(\Q(\zeta_{17})^+\), 8.0.4079249161.1 |
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: | 16.0.137350965859713069141239809.11 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(17\) | 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |