Normalized defining polynomial
\( x^{16} - 3 x^{15} + 60 x^{14} - 197 x^{13} + 1713 x^{12} - 4867 x^{11} + 24121 x^{10} - 51334 x^{9} + \cdots + 270301 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(536385794583341500395870753\) \(\medspace = 3^{8}\cdot 13^{4}\cdot 17^{15}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(46.84\) | 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: | $3^{1/2}13^{1/2}17^{15/16}\approx 88.9361176789668$ | ||
Ramified primes: | \(3\), \(13\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{101}a^{14}-\frac{11}{101}a^{13}+\frac{48}{101}a^{12}+\frac{14}{101}a^{11}+\frac{33}{101}a^{10}+\frac{34}{101}a^{9}+\frac{46}{101}a^{8}+\frac{44}{101}a^{7}-\frac{38}{101}a^{6}-\frac{22}{101}a^{5}-\frac{26}{101}a^{4}+\frac{4}{101}a^{3}+\frac{42}{101}a^{2}+\frac{11}{101}a-\frac{25}{101}$, $\frac{1}{12\!\cdots\!23}a^{15}+\frac{36\!\cdots\!67}{12\!\cdots\!23}a^{14}+\frac{11\!\cdots\!86}{12\!\cdots\!23}a^{13}+\frac{48\!\cdots\!27}{12\!\cdots\!23}a^{12}-\frac{24\!\cdots\!13}{12\!\cdots\!23}a^{11}-\frac{27\!\cdots\!79}{12\!\cdots\!23}a^{10}-\frac{20\!\cdots\!27}{12\!\cdots\!23}a^{9}-\frac{12\!\cdots\!00}{12\!\cdots\!23}a^{8}-\frac{50\!\cdots\!38}{12\!\cdots\!23}a^{7}-\frac{38\!\cdots\!54}{12\!\cdots\!23}a^{6}+\frac{79\!\cdots\!99}{12\!\cdots\!23}a^{5}-\frac{34\!\cdots\!15}{12\!\cdots\!23}a^{4}+\frac{32\!\cdots\!11}{12\!\cdots\!23}a^{3}-\frac{18\!\cdots\!89}{12\!\cdots\!23}a^{2}-\frac{39\!\cdots\!88}{12\!\cdots\!23}a-\frac{30\!\cdots\!00}{12\!\cdots\!23}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{22}\times C_{22}$, which has order $968$ (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: | $\frac{79\!\cdots\!02}{12\!\cdots\!23}a^{15}-\frac{20\!\cdots\!18}{12\!\cdots\!23}a^{14}+\frac{45\!\cdots\!27}{12\!\cdots\!23}a^{13}-\frac{13\!\cdots\!90}{12\!\cdots\!23}a^{12}+\frac{12\!\cdots\!20}{12\!\cdots\!23}a^{11}-\frac{32\!\cdots\!36}{12\!\cdots\!23}a^{10}+\frac{16\!\cdots\!41}{12\!\cdots\!23}a^{9}-\frac{32\!\cdots\!92}{12\!\cdots\!23}a^{8}+\frac{93\!\cdots\!50}{12\!\cdots\!23}a^{7}-\frac{12\!\cdots\!27}{12\!\cdots\!23}a^{6}+\frac{19\!\cdots\!97}{12\!\cdots\!23}a^{5}-\frac{22\!\cdots\!67}{12\!\cdots\!23}a^{4}+\frac{29\!\cdots\!73}{12\!\cdots\!23}a^{3}-\frac{22\!\cdots\!49}{12\!\cdots\!23}a^{2}+\frac{18\!\cdots\!10}{12\!\cdots\!23}a-\frac{36\!\cdots\!94}{12\!\cdots\!23}$, $\frac{87\!\cdots\!90}{12\!\cdots\!23}a^{15}-\frac{15\!\cdots\!05}{12\!\cdots\!23}a^{14}+\frac{48\!\cdots\!59}{12\!\cdots\!23}a^{13}-\frac{10\!\cdots\!61}{12\!\cdots\!23}a^{12}+\frac{12\!\cdots\!41}{12\!\cdots\!23}a^{11}-\frac{22\!\cdots\!90}{12\!\cdots\!23}a^{10}+\frac{14\!\cdots\!12}{12\!\cdots\!23}a^{9}-\frac{14\!\cdots\!22}{12\!\cdots\!23}a^{8}+\frac{59\!\cdots\!75}{12\!\cdots\!23}a^{7}-\frac{11\!\cdots\!28}{12\!\cdots\!23}a^{6}+\frac{46\!\cdots\!42}{12\!\cdots\!23}a^{5}+\frac{15\!\cdots\!01}{12\!\cdots\!23}a^{4}-\frac{46\!\cdots\!47}{12\!\cdots\!23}a^{3}+\frac{18\!\cdots\!73}{12\!\cdots\!23}a^{2}+\frac{25\!\cdots\!42}{12\!\cdots\!23}a+\frac{32\!\cdots\!04}{12\!\cdots\!23}$, $\frac{48\!\cdots\!59}{12\!\cdots\!23}a^{15}-\frac{79\!\cdots\!87}{12\!\cdots\!23}a^{14}+\frac{25\!\cdots\!87}{12\!\cdots\!23}a^{13}-\frac{13\!\cdots\!92}{12\!\cdots\!23}a^{12}+\frac{58\!\cdots\!16}{12\!\cdots\!23}a^{11}-\frac{64\!\cdots\!91}{12\!\cdots\!23}a^{10}+\frac{57\!\cdots\!13}{12\!\cdots\!23}a^{9}+\frac{68\!\cdots\!36}{12\!\cdots\!23}a^{8}+\frac{14\!\cdots\!05}{12\!\cdots\!23}a^{7}+\frac{89\!\cdots\!90}{12\!\cdots\!23}a^{6}-\frac{57\!\cdots\!94}{12\!\cdots\!23}a^{5}+\frac{29\!\cdots\!84}{12\!\cdots\!23}a^{4}-\frac{11\!\cdots\!41}{12\!\cdots\!23}a^{3}+\frac{34\!\cdots\!24}{12\!\cdots\!23}a^{2}-\frac{54\!\cdots\!50}{12\!\cdots\!23}a+\frac{32\!\cdots\!11}{12\!\cdots\!23}$, $\frac{39\!\cdots\!61}{12\!\cdots\!23}a^{15}-\frac{89\!\cdots\!23}{12\!\cdots\!23}a^{14}+\frac{22\!\cdots\!56}{12\!\cdots\!23}a^{13}-\frac{58\!\cdots\!85}{12\!\cdots\!23}a^{12}+\frac{57\!\cdots\!59}{12\!\cdots\!23}a^{11}-\frac{13\!\cdots\!59}{12\!\cdots\!23}a^{10}+\frac{67\!\cdots\!37}{12\!\cdots\!23}a^{9}-\frac{10\!\cdots\!35}{12\!\cdots\!23}a^{8}+\frac{29\!\cdots\!17}{12\!\cdots\!23}a^{7}-\frac{21\!\cdots\!84}{12\!\cdots\!23}a^{6}+\frac{62\!\cdots\!70}{12\!\cdots\!23}a^{5}+\frac{23\!\cdots\!58}{12\!\cdots\!23}a^{4}-\frac{19\!\cdots\!73}{12\!\cdots\!23}a^{3}+\frac{31\!\cdots\!37}{12\!\cdots\!23}a^{2}+\frac{43\!\cdots\!10}{12\!\cdots\!23}a+\frac{24\!\cdots\!48}{12\!\cdots\!23}$, $\frac{87\!\cdots\!90}{12\!\cdots\!23}a^{15}-\frac{15\!\cdots\!05}{12\!\cdots\!23}a^{14}+\frac{48\!\cdots\!59}{12\!\cdots\!23}a^{13}-\frac{10\!\cdots\!61}{12\!\cdots\!23}a^{12}+\frac{12\!\cdots\!41}{12\!\cdots\!23}a^{11}-\frac{22\!\cdots\!90}{12\!\cdots\!23}a^{10}+\frac{14\!\cdots\!12}{12\!\cdots\!23}a^{9}-\frac{14\!\cdots\!22}{12\!\cdots\!23}a^{8}+\frac{59\!\cdots\!75}{12\!\cdots\!23}a^{7}-\frac{11\!\cdots\!28}{12\!\cdots\!23}a^{6}+\frac{46\!\cdots\!42}{12\!\cdots\!23}a^{5}+\frac{15\!\cdots\!01}{12\!\cdots\!23}a^{4}-\frac{46\!\cdots\!47}{12\!\cdots\!23}a^{3}+\frac{18\!\cdots\!73}{12\!\cdots\!23}a^{2}+\frac{25\!\cdots\!42}{12\!\cdots\!23}a+\frac{19\!\cdots\!81}{12\!\cdots\!23}$, $\frac{40\!\cdots\!98}{12\!\cdots\!23}a^{15}-\frac{11\!\cdots\!21}{12\!\cdots\!23}a^{14}+\frac{22\!\cdots\!78}{12\!\cdots\!23}a^{13}-\frac{73\!\cdots\!02}{12\!\cdots\!23}a^{12}+\frac{60\!\cdots\!55}{12\!\cdots\!23}a^{11}-\frac{17\!\cdots\!38}{12\!\cdots\!23}a^{10}+\frac{76\!\cdots\!68}{12\!\cdots\!23}a^{9}-\frac{15\!\cdots\!68}{12\!\cdots\!23}a^{8}+\frac{38\!\cdots\!56}{12\!\cdots\!23}a^{7}-\frac{56\!\cdots\!68}{12\!\cdots\!23}a^{6}+\frac{42\!\cdots\!70}{12\!\cdots\!23}a^{5}-\frac{69\!\cdots\!60}{12\!\cdots\!23}a^{4}+\frac{42\!\cdots\!02}{12\!\cdots\!23}a^{3}-\frac{70\!\cdots\!81}{12\!\cdots\!23}a^{2}+\frac{38\!\cdots\!06}{12\!\cdots\!23}a-\frac{55\!\cdots\!47}{12\!\cdots\!23}$, $\frac{29\!\cdots\!51}{12\!\cdots\!23}a^{15}-\frac{69\!\cdots\!90}{12\!\cdots\!23}a^{14}+\frac{16\!\cdots\!16}{12\!\cdots\!23}a^{13}-\frac{46\!\cdots\!28}{12\!\cdots\!23}a^{12}+\frac{44\!\cdots\!56}{12\!\cdots\!23}a^{11}-\frac{10\!\cdots\!93}{12\!\cdots\!23}a^{10}+\frac{56\!\cdots\!36}{12\!\cdots\!23}a^{9}-\frac{98\!\cdots\!55}{12\!\cdots\!23}a^{8}+\frac{30\!\cdots\!30}{12\!\cdots\!23}a^{7}-\frac{34\!\cdots\!50}{12\!\cdots\!23}a^{6}+\frac{52\!\cdots\!53}{12\!\cdots\!23}a^{5}-\frac{46\!\cdots\!78}{12\!\cdots\!23}a^{4}+\frac{76\!\cdots\!60}{12\!\cdots\!23}a^{3}-\frac{42\!\cdots\!59}{12\!\cdots\!23}a^{2}+\frac{49\!\cdots\!20}{12\!\cdots\!23}a-\frac{11\!\cdots\!53}{12\!\cdots\!23}$ (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
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 968}{2\cdot\sqrt{536385794583341500395870753}}\cr\approx \mathstrut & 0.184777256899 \end{aligned}\] (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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | $16$ | $16$ | $16$ | R | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.16.8.2 | $x^{16} - 54 x^{10} + 81 x^{8} + 1458 x^{4} - 4374 x^{2} + 13122$ | $2$ | $8$ | $8$ | $C_{16}$ | $[\ ]_{2}^{8}$ |
\(13\) | 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(17\) | 17.16.15.1 | $x^{16} + 272$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |