Normalized defining polynomial
\( x^{16} - x^{15} + 35 x^{14} - 52 x^{13} + 426 x^{12} - 834 x^{11} + 2908 x^{10} - 5441 x^{9} + 12734 x^{8} - 20894 x^{7} + 36823 x^{6} - 49114 x^{5} + 65638 x^{4} + \cdots + 19211 \)
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: | \(22585894701900222828166433\) \(\medspace = 17^{15}\cdot 53^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.43\) | 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: | $17^{15/16}53^{1/2}\approx 103.67732863884214$ | ||
Ramified primes: | \(17\), \(53\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{16013}a^{14}-\frac{2829}{16013}a^{13}-\frac{970}{16013}a^{12}+\frac{4752}{16013}a^{11}-\frac{1829}{16013}a^{10}+\frac{6191}{16013}a^{9}+\frac{3715}{16013}a^{8}-\frac{3625}{16013}a^{7}+\frac{3919}{16013}a^{6}-\frac{1729}{16013}a^{5}-\frac{5464}{16013}a^{4}+\frac{1202}{16013}a^{3}+\frac{6129}{16013}a^{2}-\frac{199}{16013}a-\frac{5898}{16013}$, $\frac{1}{69\!\cdots\!91}a^{15}+\frac{14\!\cdots\!10}{69\!\cdots\!91}a^{14}-\frac{20\!\cdots\!73}{10\!\cdots\!73}a^{13}-\frac{36\!\cdots\!47}{69\!\cdots\!91}a^{12}+\frac{11\!\cdots\!72}{69\!\cdots\!91}a^{11}+\frac{10\!\cdots\!86}{69\!\cdots\!91}a^{10}-\frac{41\!\cdots\!19}{69\!\cdots\!91}a^{9}-\frac{25\!\cdots\!74}{69\!\cdots\!91}a^{8}+\frac{23\!\cdots\!96}{69\!\cdots\!91}a^{7}-\frac{88\!\cdots\!48}{28\!\cdots\!69}a^{6}+\frac{11\!\cdots\!45}{69\!\cdots\!91}a^{5}-\frac{57\!\cdots\!22}{69\!\cdots\!91}a^{4}+\frac{15\!\cdots\!61}{69\!\cdots\!91}a^{3}+\frac{42\!\cdots\!95}{69\!\cdots\!91}a^{2}-\frac{32\!\cdots\!04}{69\!\cdots\!91}a-\frac{31\!\cdots\!72}{69\!\cdots\!91}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{9}\times C_{18}$, which has order $162$ (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: | $\frac{13\!\cdots\!18}{69\!\cdots\!91}a^{15}+\frac{41\!\cdots\!85}{69\!\cdots\!91}a^{14}+\frac{70\!\cdots\!02}{10\!\cdots\!73}a^{13}-\frac{54\!\cdots\!28}{69\!\cdots\!91}a^{12}+\frac{51\!\cdots\!97}{69\!\cdots\!91}a^{11}-\frac{27\!\cdots\!80}{69\!\cdots\!91}a^{10}+\frac{27\!\cdots\!66}{69\!\cdots\!91}a^{9}-\frac{57\!\cdots\!65}{28\!\cdots\!69}a^{8}+\frac{88\!\cdots\!19}{69\!\cdots\!91}a^{7}-\frac{32\!\cdots\!37}{69\!\cdots\!91}a^{6}+\frac{16\!\cdots\!31}{69\!\cdots\!91}a^{5}+\frac{72\!\cdots\!34}{69\!\cdots\!91}a^{4}+\frac{12\!\cdots\!21}{69\!\cdots\!91}a^{3}+\frac{13\!\cdots\!65}{69\!\cdots\!91}a^{2}-\frac{13\!\cdots\!94}{69\!\cdots\!91}a+\frac{22\!\cdots\!84}{69\!\cdots\!91}$, $\frac{28\!\cdots\!35}{69\!\cdots\!91}a^{15}-\frac{42\!\cdots\!07}{69\!\cdots\!91}a^{14}+\frac{89\!\cdots\!30}{69\!\cdots\!91}a^{13}-\frac{19\!\cdots\!84}{69\!\cdots\!91}a^{12}+\frac{92\!\cdots\!60}{69\!\cdots\!91}a^{11}-\frac{27\!\cdots\!03}{69\!\cdots\!91}a^{10}+\frac{55\!\cdots\!65}{69\!\cdots\!91}a^{9}-\frac{14\!\cdots\!59}{69\!\cdots\!91}a^{8}+\frac{22\!\cdots\!54}{69\!\cdots\!91}a^{7}-\frac{47\!\cdots\!80}{69\!\cdots\!91}a^{6}+\frac{57\!\cdots\!09}{69\!\cdots\!91}a^{5}-\frac{91\!\cdots\!10}{69\!\cdots\!91}a^{4}+\frac{79\!\cdots\!66}{69\!\cdots\!91}a^{3}-\frac{94\!\cdots\!14}{69\!\cdots\!91}a^{2}+\frac{52\!\cdots\!27}{69\!\cdots\!91}a-\frac{47\!\cdots\!68}{69\!\cdots\!91}$, $\frac{95\!\cdots\!10}{69\!\cdots\!91}a^{15}-\frac{61\!\cdots\!57}{69\!\cdots\!91}a^{14}+\frac{32\!\cdots\!11}{69\!\cdots\!91}a^{13}-\frac{22\!\cdots\!72}{69\!\cdots\!91}a^{12}+\frac{47\!\cdots\!79}{69\!\cdots\!91}a^{11}-\frac{26\!\cdots\!71}{69\!\cdots\!91}a^{10}+\frac{46\!\cdots\!01}{69\!\cdots\!91}a^{9}-\frac{15\!\cdots\!94}{69\!\cdots\!91}a^{8}+\frac{23\!\cdots\!01}{69\!\cdots\!91}a^{7}-\frac{54\!\cdots\!86}{69\!\cdots\!91}a^{6}+\frac{73\!\cdots\!27}{69\!\cdots\!91}a^{5}-\frac{12\!\cdots\!54}{69\!\cdots\!91}a^{4}+\frac{12\!\cdots\!96}{69\!\cdots\!91}a^{3}-\frac{15\!\cdots\!78}{69\!\cdots\!91}a^{2}+\frac{10\!\cdots\!84}{69\!\cdots\!91}a-\frac{70\!\cdots\!56}{69\!\cdots\!91}$, $\frac{27\!\cdots\!74}{69\!\cdots\!91}a^{15}+\frac{31\!\cdots\!90}{69\!\cdots\!91}a^{14}+\frac{88\!\cdots\!83}{69\!\cdots\!91}a^{13}-\frac{47\!\cdots\!65}{69\!\cdots\!91}a^{12}+\frac{87\!\cdots\!89}{69\!\cdots\!91}a^{11}-\frac{10\!\cdots\!34}{69\!\cdots\!91}a^{10}+\frac{42\!\cdots\!97}{69\!\cdots\!91}a^{9}-\frac{56\!\cdots\!28}{69\!\cdots\!91}a^{8}+\frac{13\!\cdots\!25}{69\!\cdots\!91}a^{7}-\frac{14\!\cdots\!40}{69\!\cdots\!91}a^{6}+\frac{24\!\cdots\!48}{69\!\cdots\!91}a^{5}-\frac{16\!\cdots\!87}{69\!\cdots\!91}a^{4}+\frac{14\!\cdots\!28}{69\!\cdots\!91}a^{3}+\frac{55\!\cdots\!20}{69\!\cdots\!91}a^{2}-\frac{71\!\cdots\!83}{69\!\cdots\!91}a+\frac{16\!\cdots\!05}{69\!\cdots\!91}$, $\frac{63\!\cdots\!61}{69\!\cdots\!91}a^{15}-\frac{42\!\cdots\!97}{69\!\cdots\!91}a^{14}+\frac{79\!\cdots\!47}{69\!\cdots\!91}a^{13}-\frac{14\!\cdots\!19}{69\!\cdots\!91}a^{12}+\frac{48\!\cdots\!71}{69\!\cdots\!91}a^{11}-\frac{16\!\cdots\!69}{69\!\cdots\!91}a^{10}+\frac{20\!\cdots\!04}{10\!\cdots\!73}a^{9}-\frac{93\!\cdots\!31}{69\!\cdots\!91}a^{8}+\frac{89\!\cdots\!29}{69\!\cdots\!91}a^{7}-\frac{33\!\cdots\!40}{69\!\cdots\!91}a^{6}+\frac{33\!\cdots\!61}{69\!\cdots\!91}a^{5}-\frac{74\!\cdots\!23}{69\!\cdots\!91}a^{4}+\frac{64\!\cdots\!38}{69\!\cdots\!91}a^{3}-\frac{10\!\cdots\!34}{69\!\cdots\!91}a^{2}+\frac{59\!\cdots\!10}{69\!\cdots\!91}a-\frac{57\!\cdots\!82}{69\!\cdots\!91}$, $\frac{20\!\cdots\!06}{69\!\cdots\!91}a^{15}-\frac{19\!\cdots\!31}{69\!\cdots\!91}a^{14}+\frac{64\!\cdots\!56}{69\!\cdots\!91}a^{13}-\frac{10\!\cdots\!71}{69\!\cdots\!91}a^{12}+\frac{66\!\cdots\!08}{69\!\cdots\!91}a^{11}-\frac{17\!\cdots\!79}{69\!\cdots\!91}a^{10}+\frac{37\!\cdots\!12}{69\!\cdots\!91}a^{9}-\frac{98\!\cdots\!00}{69\!\cdots\!91}a^{8}+\frac{15\!\cdots\!73}{69\!\cdots\!91}a^{7}-\frac{32\!\cdots\!11}{69\!\cdots\!91}a^{6}+\frac{41\!\cdots\!01}{69\!\cdots\!91}a^{5}-\frac{66\!\cdots\!76}{69\!\cdots\!91}a^{4}+\frac{64\!\cdots\!13}{69\!\cdots\!91}a^{3}-\frac{75\!\cdots\!61}{69\!\cdots\!91}a^{2}+\frac{45\!\cdots\!51}{69\!\cdots\!91}a-\frac{33\!\cdots\!21}{69\!\cdots\!91}$, $\frac{13\!\cdots\!84}{69\!\cdots\!91}a^{15}-\frac{35\!\cdots\!25}{69\!\cdots\!91}a^{14}+\frac{44\!\cdots\!33}{69\!\cdots\!91}a^{13}-\frac{14\!\cdots\!37}{69\!\cdots\!91}a^{12}+\frac{52\!\cdots\!71}{69\!\cdots\!91}a^{11}-\frac{20\!\cdots\!63}{69\!\cdots\!91}a^{10}+\frac{39\!\cdots\!20}{69\!\cdots\!91}a^{9}-\frac{12\!\cdots\!96}{69\!\cdots\!91}a^{8}+\frac{19\!\cdots\!75}{69\!\cdots\!91}a^{7}-\frac{43\!\cdots\!92}{69\!\cdots\!91}a^{6}+\frac{57\!\cdots\!90}{69\!\cdots\!91}a^{5}-\frac{94\!\cdots\!98}{69\!\cdots\!91}a^{4}+\frac{97\!\cdots\!31}{69\!\cdots\!91}a^{3}-\frac{11\!\cdots\!48}{69\!\cdots\!91}a^{2}+\frac{76\!\cdots\!57}{69\!\cdots\!91}a-\frac{56\!\cdots\!25}{69\!\cdots\!91}$ (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: | \( 3640.01221338 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 162}{2\cdot\sqrt{22585894701900222828166433}}\cr\approx \mathstrut & 0.150698231956 \end{aligned}\] (assuming GRH)
Galois group
$\OD_{32}$ (as 16T22):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_{16} : C_2$ |
Character table for $C_{16} : C_2$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.16.15.1 | $x^{16} + 272$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
\(53\) | 53.8.4.2 | $x^{8} + 25281 x^{4} - 5657326 x^{2} + 15780962$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
53.8.0.1 | $x^{8} + 8 x^{4} + 29 x^{3} + 18 x^{2} + x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |