Normalized defining polynomial
\( x^{16} - 2 x^{15} - 13 x^{14} + 26 x^{13} - x^{12} + 19 x^{11} + 370 x^{10} - 332 x^{9} - 1359 x^{8} + \cdots + 52 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(57681033264163530732453953\) \(\medspace = 17^{15}\cdot 67^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(40.74\) | 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: | $17^{15/16}67^{1/2}\approx 116.56905215178652$ | ||
Ramified primes: | \(17\), \(67\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{1}{8}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{3}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{22\!\cdots\!28}a^{15}+\frac{13\!\cdots\!43}{22\!\cdots\!28}a^{14}+\frac{12\!\cdots\!13}{56\!\cdots\!32}a^{13}+\frac{27\!\cdots\!23}{56\!\cdots\!32}a^{12}-\frac{68\!\cdots\!57}{22\!\cdots\!28}a^{11}+\frac{23\!\cdots\!35}{11\!\cdots\!64}a^{10}-\frac{36\!\cdots\!21}{11\!\cdots\!64}a^{9}+\frac{14\!\cdots\!87}{11\!\cdots\!64}a^{8}-\frac{61\!\cdots\!85}{22\!\cdots\!28}a^{7}+\frac{14\!\cdots\!51}{22\!\cdots\!28}a^{6}-\frac{39\!\cdots\!69}{22\!\cdots\!28}a^{5}+\frac{19\!\cdots\!29}{22\!\cdots\!28}a^{4}+\frac{38\!\cdots\!45}{11\!\cdots\!64}a^{3}-\frac{62\!\cdots\!03}{22\!\cdots\!28}a^{2}+\frac{39\!\cdots\!55}{11\!\cdots\!64}a+\frac{87\!\cdots\!69}{14\!\cdots\!33}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $9$ | 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: | $\frac{11\!\cdots\!04}{37\!\cdots\!21}a^{15}+\frac{14\!\cdots\!21}{30\!\cdots\!68}a^{14}-\frac{18\!\cdots\!07}{30\!\cdots\!68}a^{13}-\frac{51\!\cdots\!55}{75\!\cdots\!42}a^{12}+\frac{21\!\cdots\!19}{75\!\cdots\!42}a^{11}+\frac{48\!\cdots\!71}{30\!\cdots\!68}a^{10}+\frac{94\!\cdots\!65}{75\!\cdots\!42}a^{9}+\frac{10\!\cdots\!07}{37\!\cdots\!21}a^{8}-\frac{57\!\cdots\!39}{75\!\cdots\!42}a^{7}-\frac{60\!\cdots\!47}{30\!\cdots\!68}a^{6}-\frac{47\!\cdots\!05}{30\!\cdots\!68}a^{5}-\frac{44\!\cdots\!91}{30\!\cdots\!68}a^{4}-\frac{24\!\cdots\!75}{30\!\cdots\!68}a^{3}-\frac{98\!\cdots\!19}{37\!\cdots\!21}a^{2}-\frac{86\!\cdots\!53}{30\!\cdots\!68}a+\frac{93\!\cdots\!25}{15\!\cdots\!84}$, $\frac{22\!\cdots\!95}{60\!\cdots\!36}a^{15}-\frac{51\!\cdots\!37}{60\!\cdots\!36}a^{14}-\frac{13\!\cdots\!61}{30\!\cdots\!68}a^{13}+\frac{16\!\cdots\!59}{15\!\cdots\!84}a^{12}-\frac{14\!\cdots\!47}{60\!\cdots\!36}a^{11}+\frac{16\!\cdots\!47}{15\!\cdots\!84}a^{10}+\frac{38\!\cdots\!55}{30\!\cdots\!68}a^{9}-\frac{50\!\cdots\!25}{30\!\cdots\!68}a^{8}-\frac{29\!\cdots\!55}{60\!\cdots\!36}a^{7}-\frac{15\!\cdots\!01}{60\!\cdots\!36}a^{6}-\frac{27\!\cdots\!69}{60\!\cdots\!36}a^{5}+\frac{16\!\cdots\!09}{60\!\cdots\!36}a^{4}+\frac{38\!\cdots\!47}{15\!\cdots\!84}a^{3}+\frac{29\!\cdots\!95}{60\!\cdots\!36}a^{2}+\frac{15\!\cdots\!43}{75\!\cdots\!42}a+\frac{21\!\cdots\!13}{15\!\cdots\!84}$, $\frac{47\!\cdots\!53}{60\!\cdots\!36}a^{15}-\frac{18\!\cdots\!61}{60\!\cdots\!36}a^{14}-\frac{60\!\cdots\!35}{75\!\cdots\!42}a^{13}+\frac{64\!\cdots\!85}{15\!\cdots\!84}a^{12}-\frac{19\!\cdots\!57}{60\!\cdots\!36}a^{11}-\frac{53\!\cdots\!51}{30\!\cdots\!68}a^{10}+\frac{87\!\cdots\!91}{30\!\cdots\!68}a^{9}-\frac{24\!\cdots\!05}{30\!\cdots\!68}a^{8}-\frac{49\!\cdots\!01}{60\!\cdots\!36}a^{7}+\frac{12\!\cdots\!95}{60\!\cdots\!36}a^{6}+\frac{31\!\cdots\!27}{60\!\cdots\!36}a^{5}+\frac{39\!\cdots\!21}{60\!\cdots\!36}a^{4}+\frac{30\!\cdots\!07}{30\!\cdots\!68}a^{3}+\frac{10\!\cdots\!85}{60\!\cdots\!36}a^{2}-\frac{10\!\cdots\!59}{30\!\cdots\!68}a-\frac{14\!\cdots\!21}{75\!\cdots\!42}$, $\frac{87\!\cdots\!85}{22\!\cdots\!28}a^{15}-\frac{82\!\cdots\!73}{22\!\cdots\!28}a^{14}-\frac{88\!\cdots\!74}{14\!\cdots\!33}a^{13}+\frac{29\!\cdots\!99}{56\!\cdots\!32}a^{12}+\frac{37\!\cdots\!83}{22\!\cdots\!28}a^{11}+\frac{13\!\cdots\!39}{11\!\cdots\!64}a^{10}+\frac{15\!\cdots\!17}{11\!\cdots\!64}a^{9}+\frac{19\!\cdots\!01}{11\!\cdots\!64}a^{8}-\frac{18\!\cdots\!85}{22\!\cdots\!28}a^{7}-\frac{20\!\cdots\!85}{22\!\cdots\!28}a^{6}-\frac{87\!\cdots\!89}{22\!\cdots\!28}a^{5}-\frac{90\!\cdots\!15}{22\!\cdots\!28}a^{4}+\frac{52\!\cdots\!33}{11\!\cdots\!64}a^{3}+\frac{45\!\cdots\!05}{22\!\cdots\!28}a^{2}+\frac{84\!\cdots\!29}{11\!\cdots\!64}a+\frac{23\!\cdots\!79}{28\!\cdots\!66}$, $\frac{77\!\cdots\!43}{56\!\cdots\!32}a^{15}-\frac{11\!\cdots\!81}{28\!\cdots\!66}a^{14}-\frac{85\!\cdots\!49}{56\!\cdots\!32}a^{13}+\frac{14\!\cdots\!53}{28\!\cdots\!66}a^{12}-\frac{20\!\cdots\!17}{56\!\cdots\!32}a^{11}+\frac{18\!\cdots\!99}{56\!\cdots\!32}a^{10}+\frac{13\!\cdots\!35}{28\!\cdots\!66}a^{9}-\frac{12\!\cdots\!23}{14\!\cdots\!33}a^{8}-\frac{77\!\cdots\!03}{56\!\cdots\!32}a^{7}+\frac{51\!\cdots\!87}{14\!\cdots\!33}a^{6}-\frac{59\!\cdots\!53}{28\!\cdots\!66}a^{5}+\frac{16\!\cdots\!86}{14\!\cdots\!33}a^{4}-\frac{88\!\cdots\!97}{56\!\cdots\!32}a^{3}-\frac{15\!\cdots\!75}{56\!\cdots\!32}a^{2}+\frac{15\!\cdots\!73}{56\!\cdots\!32}a+\frac{10\!\cdots\!75}{28\!\cdots\!66}$, $\frac{49\!\cdots\!13}{22\!\cdots\!28}a^{15}-\frac{91\!\cdots\!35}{22\!\cdots\!28}a^{14}-\frac{29\!\cdots\!83}{11\!\cdots\!64}a^{13}+\frac{27\!\cdots\!09}{56\!\cdots\!32}a^{12}-\frac{92\!\cdots\!93}{22\!\cdots\!28}a^{11}+\frac{24\!\cdots\!97}{28\!\cdots\!66}a^{10}+\frac{99\!\cdots\!27}{11\!\cdots\!64}a^{9}-\frac{60\!\cdots\!05}{11\!\cdots\!64}a^{8}-\frac{43\!\cdots\!73}{22\!\cdots\!28}a^{7}-\frac{51\!\cdots\!35}{22\!\cdots\!28}a^{6}-\frac{22\!\cdots\!03}{22\!\cdots\!28}a^{5}-\frac{23\!\cdots\!77}{22\!\cdots\!28}a^{4}-\frac{26\!\cdots\!21}{28\!\cdots\!66}a^{3}-\frac{13\!\cdots\!07}{22\!\cdots\!28}a^{2}-\frac{71\!\cdots\!07}{28\!\cdots\!66}a-\frac{36\!\cdots\!01}{56\!\cdots\!32}$, $\frac{87\!\cdots\!85}{22\!\cdots\!28}a^{15}-\frac{82\!\cdots\!73}{22\!\cdots\!28}a^{14}-\frac{88\!\cdots\!74}{14\!\cdots\!33}a^{13}+\frac{29\!\cdots\!99}{56\!\cdots\!32}a^{12}+\frac{37\!\cdots\!83}{22\!\cdots\!28}a^{11}+\frac{13\!\cdots\!39}{11\!\cdots\!64}a^{10}+\frac{15\!\cdots\!17}{11\!\cdots\!64}a^{9}+\frac{19\!\cdots\!01}{11\!\cdots\!64}a^{8}-\frac{18\!\cdots\!85}{22\!\cdots\!28}a^{7}-\frac{20\!\cdots\!85}{22\!\cdots\!28}a^{6}-\frac{87\!\cdots\!89}{22\!\cdots\!28}a^{5}-\frac{90\!\cdots\!15}{22\!\cdots\!28}a^{4}+\frac{52\!\cdots\!33}{11\!\cdots\!64}a^{3}+\frac{45\!\cdots\!05}{22\!\cdots\!28}a^{2}-\frac{27\!\cdots\!35}{11\!\cdots\!64}a-\frac{47\!\cdots\!87}{28\!\cdots\!66}$, $\frac{13\!\cdots\!61}{22\!\cdots\!28}a^{15}-\frac{38\!\cdots\!57}{22\!\cdots\!28}a^{14}-\frac{38\!\cdots\!93}{56\!\cdots\!32}a^{13}+\frac{11\!\cdots\!59}{56\!\cdots\!32}a^{12}-\frac{36\!\cdots\!49}{22\!\cdots\!28}a^{11}+\frac{24\!\cdots\!77}{11\!\cdots\!64}a^{10}+\frac{23\!\cdots\!09}{11\!\cdots\!64}a^{9}-\frac{41\!\cdots\!51}{11\!\cdots\!64}a^{8}-\frac{13\!\cdots\!37}{22\!\cdots\!28}a^{7}-\frac{11\!\cdots\!81}{22\!\cdots\!28}a^{6}-\frac{24\!\cdots\!73}{22\!\cdots\!28}a^{5}+\frac{14\!\cdots\!25}{22\!\cdots\!28}a^{4}-\frac{50\!\cdots\!21}{11\!\cdots\!64}a^{3}+\frac{70\!\cdots\!65}{22\!\cdots\!28}a^{2}-\frac{68\!\cdots\!85}{11\!\cdots\!64}a+\frac{62\!\cdots\!52}{14\!\cdots\!33}$, $\frac{83\!\cdots\!07}{56\!\cdots\!32}a^{15}-\frac{48\!\cdots\!45}{11\!\cdots\!64}a^{14}-\frac{18\!\cdots\!89}{11\!\cdots\!64}a^{13}+\frac{15\!\cdots\!97}{28\!\cdots\!66}a^{12}-\frac{23\!\cdots\!33}{56\!\cdots\!32}a^{11}+\frac{44\!\cdots\!07}{11\!\cdots\!64}a^{10}+\frac{28\!\cdots\!93}{56\!\cdots\!32}a^{9}-\frac{53\!\cdots\!15}{56\!\cdots\!32}a^{8}-\frac{19\!\cdots\!73}{14\!\cdots\!33}a^{7}+\frac{32\!\cdots\!97}{11\!\cdots\!64}a^{6}-\frac{24\!\cdots\!83}{11\!\cdots\!64}a^{5}+\frac{75\!\cdots\!41}{11\!\cdots\!64}a^{4}-\frac{73\!\cdots\!95}{11\!\cdots\!64}a^{3}-\frac{55\!\cdots\!36}{14\!\cdots\!33}a^{2}+\frac{44\!\cdots\!73}{11\!\cdots\!64}a-\frac{22\!\cdots\!13}{56\!\cdots\!32}$ (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: | \( 3266009.09307 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 3266009.09307 \cdot 4}{2\cdot\sqrt{57681033264163530732453953}}\cr\approx \mathstrut & 0.846700979680 \end{aligned}\] (assuming GRH)
Galois group
$C_2^7.C_8$ (as 16T1155):
A solvable group of order 1024 |
The 40 conjugacy class representatives for $C_2^7.C_8$ |
Character table for $C_2^7.C_8$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.1842010303097.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | 16.4.57681033264163530732453953.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
\(67\) | $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |