Normalized defining polynomial
\( x^{16} - 2 x^{15} - 75 x^{14} - 28 x^{13} + 2087 x^{12} + 12539 x^{11} - 23154 x^{10} - 528593 x^{9} + \cdots + 10314571 \)
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: | \(21524562403589040109288671558095801\) \(\medspace = 7^{12}\cdot 41^{15}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(139.90\) | 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: | $7^{3/4}41^{15/16}\approx 139.8969981496878$ | ||
Ramified primes: | \(7\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{50\!\cdots\!27}a^{15}-\frac{22\!\cdots\!92}{50\!\cdots\!27}a^{14}+\frac{11\!\cdots\!65}{50\!\cdots\!27}a^{13}-\frac{30\!\cdots\!84}{50\!\cdots\!27}a^{12}-\frac{11\!\cdots\!15}{50\!\cdots\!27}a^{11}+\frac{19\!\cdots\!32}{50\!\cdots\!27}a^{10}+\frac{20\!\cdots\!46}{50\!\cdots\!27}a^{9}+\frac{39\!\cdots\!44}{50\!\cdots\!27}a^{8}+\frac{24\!\cdots\!06}{50\!\cdots\!27}a^{7}-\frac{36\!\cdots\!57}{50\!\cdots\!27}a^{6}+\frac{11\!\cdots\!07}{50\!\cdots\!27}a^{5}+\frac{18\!\cdots\!12}{50\!\cdots\!27}a^{4}+\frac{16\!\cdots\!25}{50\!\cdots\!27}a^{3}+\frac{20\!\cdots\!23}{50\!\cdots\!27}a^{2}+\frac{19\!\cdots\!77}{50\!\cdots\!27}a-\frac{21\!\cdots\!77}{50\!\cdots\!27}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | $\frac{10\!\cdots\!47}{21\!\cdots\!99}a^{15}-\frac{34\!\cdots\!86}{21\!\cdots\!99}a^{14}-\frac{71\!\cdots\!22}{21\!\cdots\!99}a^{13}+\frac{65\!\cdots\!19}{21\!\cdots\!99}a^{12}+\frac{20\!\cdots\!20}{21\!\cdots\!99}a^{11}+\frac{10\!\cdots\!78}{21\!\cdots\!99}a^{10}-\frac{36\!\cdots\!71}{21\!\cdots\!99}a^{9}-\frac{48\!\cdots\!96}{21\!\cdots\!99}a^{8}+\frac{57\!\cdots\!86}{21\!\cdots\!99}a^{7}+\frac{81\!\cdots\!44}{21\!\cdots\!99}a^{6}-\frac{67\!\cdots\!18}{21\!\cdots\!99}a^{5}-\frac{53\!\cdots\!03}{21\!\cdots\!99}a^{4}+\frac{38\!\cdots\!67}{21\!\cdots\!99}a^{3}+\frac{97\!\cdots\!36}{21\!\cdots\!99}a^{2}-\frac{63\!\cdots\!67}{21\!\cdots\!99}a+\frac{17\!\cdots\!21}{21\!\cdots\!99}$, $\frac{23\!\cdots\!68}{21\!\cdots\!99}a^{15}-\frac{84\!\cdots\!55}{21\!\cdots\!99}a^{14}-\frac{15\!\cdots\!41}{21\!\cdots\!99}a^{13}+\frac{20\!\cdots\!08}{21\!\cdots\!99}a^{12}+\frac{44\!\cdots\!43}{21\!\cdots\!99}a^{11}+\frac{21\!\cdots\!59}{21\!\cdots\!99}a^{10}-\frac{87\!\cdots\!58}{21\!\cdots\!99}a^{9}-\frac{10\!\cdots\!53}{21\!\cdots\!99}a^{8}+\frac{16\!\cdots\!51}{21\!\cdots\!99}a^{7}+\frac{17\!\cdots\!52}{21\!\cdots\!99}a^{6}-\frac{23\!\cdots\!61}{21\!\cdots\!99}a^{5}-\frac{10\!\cdots\!47}{21\!\cdots\!99}a^{4}+\frac{15\!\cdots\!87}{21\!\cdots\!99}a^{3}+\frac{15\!\cdots\!59}{21\!\cdots\!99}a^{2}-\frac{32\!\cdots\!49}{21\!\cdots\!99}a+\frac{17\!\cdots\!20}{21\!\cdots\!99}$, $\frac{90\!\cdots\!51}{13\!\cdots\!71}a^{15}-\frac{30\!\cdots\!81}{13\!\cdots\!71}a^{14}-\frac{64\!\cdots\!81}{13\!\cdots\!71}a^{13}+\frac{56\!\cdots\!25}{13\!\cdots\!71}a^{12}+\frac{18\!\cdots\!48}{13\!\cdots\!71}a^{11}+\frac{92\!\cdots\!08}{13\!\cdots\!71}a^{10}-\frac{33\!\cdots\!46}{13\!\cdots\!71}a^{9}-\frac{44\!\cdots\!03}{13\!\cdots\!71}a^{8}+\frac{49\!\cdots\!28}{13\!\cdots\!71}a^{7}+\frac{76\!\cdots\!83}{13\!\cdots\!71}a^{6}-\frac{50\!\cdots\!78}{13\!\cdots\!71}a^{5}-\frac{53\!\cdots\!81}{13\!\cdots\!71}a^{4}+\frac{27\!\cdots\!03}{13\!\cdots\!71}a^{3}+\frac{12\!\cdots\!36}{13\!\cdots\!71}a^{2}-\frac{78\!\cdots\!35}{13\!\cdots\!71}a-\frac{11\!\cdots\!79}{13\!\cdots\!71}$, $\frac{19\!\cdots\!56}{13\!\cdots\!71}a^{15}-\frac{78\!\cdots\!09}{13\!\cdots\!71}a^{14}-\frac{12\!\cdots\!97}{13\!\cdots\!71}a^{13}+\frac{17\!\cdots\!90}{13\!\cdots\!71}a^{12}+\frac{37\!\cdots\!56}{13\!\cdots\!71}a^{11}+\frac{19\!\cdots\!99}{13\!\cdots\!71}a^{10}-\frac{80\!\cdots\!23}{13\!\cdots\!71}a^{9}-\frac{90\!\cdots\!44}{13\!\cdots\!71}a^{8}+\frac{12\!\cdots\!98}{13\!\cdots\!71}a^{7}+\frac{14\!\cdots\!56}{13\!\cdots\!71}a^{6}-\frac{64\!\cdots\!31}{13\!\cdots\!71}a^{5}-\frac{96\!\cdots\!90}{13\!\cdots\!71}a^{4}-\frac{36\!\cdots\!60}{13\!\cdots\!71}a^{3}+\frac{27\!\cdots\!72}{13\!\cdots\!71}a^{2}+\frac{12\!\cdots\!18}{13\!\cdots\!71}a-\frac{35\!\cdots\!11}{13\!\cdots\!71}$, $\frac{23\!\cdots\!28}{21\!\cdots\!99}a^{15}-\frac{15\!\cdots\!70}{21\!\cdots\!99}a^{14}-\frac{11\!\cdots\!72}{21\!\cdots\!99}a^{13}+\frac{55\!\cdots\!26}{21\!\cdots\!99}a^{12}+\frac{30\!\cdots\!53}{21\!\cdots\!99}a^{11}+\frac{12\!\cdots\!27}{21\!\cdots\!99}a^{10}-\frac{13\!\cdots\!29}{21\!\cdots\!99}a^{9}-\frac{70\!\cdots\!51}{21\!\cdots\!99}a^{8}+\frac{38\!\cdots\!04}{21\!\cdots\!99}a^{7}+\frac{76\!\cdots\!08}{21\!\cdots\!99}a^{6}-\frac{47\!\cdots\!91}{21\!\cdots\!99}a^{5}-\frac{13\!\cdots\!11}{21\!\cdots\!99}a^{4}+\frac{25\!\cdots\!47}{21\!\cdots\!99}a^{3}-\frac{10\!\cdots\!21}{21\!\cdots\!99}a^{2}-\frac{58\!\cdots\!82}{21\!\cdots\!99}a+\frac{37\!\cdots\!13}{21\!\cdots\!99}$, $\frac{27\!\cdots\!01}{13\!\cdots\!71}a^{15}-\frac{10\!\cdots\!31}{13\!\cdots\!71}a^{14}-\frac{18\!\cdots\!78}{13\!\cdots\!71}a^{13}+\frac{26\!\cdots\!79}{13\!\cdots\!71}a^{12}+\frac{51\!\cdots\!72}{13\!\cdots\!71}a^{11}+\frac{24\!\cdots\!13}{13\!\cdots\!71}a^{10}-\frac{10\!\cdots\!42}{13\!\cdots\!71}a^{9}-\frac{12\!\cdots\!37}{13\!\cdots\!71}a^{8}+\frac{21\!\cdots\!17}{13\!\cdots\!71}a^{7}+\frac{20\!\cdots\!76}{13\!\cdots\!71}a^{6}-\frac{27\!\cdots\!19}{13\!\cdots\!71}a^{5}-\frac{12\!\cdots\!71}{13\!\cdots\!71}a^{4}+\frac{16\!\cdots\!82}{13\!\cdots\!71}a^{3}+\frac{20\!\cdots\!21}{13\!\cdots\!71}a^{2}-\frac{38\!\cdots\!92}{13\!\cdots\!71}a+\frac{24\!\cdots\!26}{13\!\cdots\!71}$, $\frac{76\!\cdots\!95}{13\!\cdots\!71}a^{15}-\frac{26\!\cdots\!76}{13\!\cdots\!71}a^{14}-\frac{55\!\cdots\!75}{13\!\cdots\!71}a^{13}+\frac{61\!\cdots\!96}{13\!\cdots\!71}a^{12}+\frac{16\!\cdots\!56}{13\!\cdots\!71}a^{11}+\frac{74\!\cdots\!18}{13\!\cdots\!71}a^{10}-\frac{30\!\cdots\!40}{13\!\cdots\!71}a^{9}-\frac{38\!\cdots\!14}{13\!\cdots\!71}a^{8}+\frac{51\!\cdots\!13}{13\!\cdots\!71}a^{7}+\frac{67\!\cdots\!26}{13\!\cdots\!71}a^{6}-\frac{64\!\cdots\!60}{13\!\cdots\!71}a^{5}-\frac{47\!\cdots\!25}{13\!\cdots\!71}a^{4}+\frac{48\!\cdots\!71}{13\!\cdots\!71}a^{3}+\frac{10\!\cdots\!83}{13\!\cdots\!71}a^{2}-\frac{16\!\cdots\!61}{13\!\cdots\!71}a+\frac{64\!\cdots\!74}{13\!\cdots\!71}$, $\frac{15\!\cdots\!76}{50\!\cdots\!27}a^{15}-\frac{29\!\cdots\!31}{50\!\cdots\!27}a^{14}-\frac{66\!\cdots\!85}{50\!\cdots\!27}a^{13}+\frac{10\!\cdots\!76}{50\!\cdots\!27}a^{12}+\frac{25\!\cdots\!36}{50\!\cdots\!27}a^{11}-\frac{62\!\cdots\!20}{50\!\cdots\!27}a^{10}-\frac{26\!\cdots\!80}{50\!\cdots\!27}a^{9}-\frac{14\!\cdots\!05}{50\!\cdots\!27}a^{8}+\frac{63\!\cdots\!21}{50\!\cdots\!27}a^{7}+\frac{11\!\cdots\!87}{50\!\cdots\!27}a^{6}-\frac{56\!\cdots\!73}{50\!\cdots\!27}a^{5}+\frac{10\!\cdots\!96}{50\!\cdots\!27}a^{4}+\frac{18\!\cdots\!99}{50\!\cdots\!27}a^{3}-\frac{11\!\cdots\!00}{50\!\cdots\!27}a^{2}-\frac{22\!\cdots\!87}{50\!\cdots\!27}a+\frac{17\!\cdots\!33}{50\!\cdots\!27}$, $\frac{27\!\cdots\!00}{50\!\cdots\!27}a^{15}+\frac{23\!\cdots\!56}{50\!\cdots\!27}a^{14}-\frac{19\!\cdots\!01}{50\!\cdots\!27}a^{13}-\frac{63\!\cdots\!71}{50\!\cdots\!27}a^{12}+\frac{38\!\cdots\!92}{50\!\cdots\!27}a^{11}+\frac{44\!\cdots\!80}{50\!\cdots\!27}a^{10}+\frac{64\!\cdots\!77}{50\!\cdots\!27}a^{9}-\frac{12\!\cdots\!38}{50\!\cdots\!27}a^{8}-\frac{36\!\cdots\!81}{50\!\cdots\!27}a^{7}+\frac{13\!\cdots\!07}{50\!\cdots\!27}a^{6}+\frac{46\!\cdots\!42}{50\!\cdots\!27}a^{5}-\frac{38\!\cdots\!34}{50\!\cdots\!27}a^{4}-\frac{13\!\cdots\!66}{50\!\cdots\!27}a^{3}+\frac{11\!\cdots\!25}{50\!\cdots\!27}a^{2}+\frac{23\!\cdots\!33}{50\!\cdots\!27}a-\frac{10\!\cdots\!54}{50\!\cdots\!27}$, $\frac{62\!\cdots\!37}{50\!\cdots\!27}a^{15}-\frac{34\!\cdots\!97}{50\!\cdots\!27}a^{14}-\frac{39\!\cdots\!16}{50\!\cdots\!27}a^{13}+\frac{13\!\cdots\!87}{50\!\cdots\!27}a^{12}+\frac{11\!\cdots\!09}{50\!\cdots\!27}a^{11}+\frac{37\!\cdots\!09}{50\!\cdots\!27}a^{10}-\frac{35\!\cdots\!60}{50\!\cdots\!27}a^{9}-\frac{25\!\cdots\!47}{50\!\cdots\!27}a^{8}+\frac{96\!\cdots\!66}{50\!\cdots\!27}a^{7}+\frac{42\!\cdots\!24}{50\!\cdots\!27}a^{6}-\frac{14\!\cdots\!21}{50\!\cdots\!27}a^{5}-\frac{22\!\cdots\!84}{50\!\cdots\!27}a^{4}+\frac{89\!\cdots\!82}{50\!\cdots\!27}a^{3}-\frac{70\!\cdots\!47}{50\!\cdots\!27}a^{2}-\frac{18\!\cdots\!44}{50\!\cdots\!27}a+\frac{14\!\cdots\!34}{50\!\cdots\!27}$, $\frac{26\!\cdots\!02}{50\!\cdots\!27}a^{15}+\frac{40\!\cdots\!42}{50\!\cdots\!27}a^{14}-\frac{16\!\cdots\!87}{50\!\cdots\!27}a^{13}-\frac{71\!\cdots\!95}{50\!\cdots\!27}a^{12}+\frac{21\!\cdots\!86}{50\!\cdots\!27}a^{11}+\frac{39\!\cdots\!21}{50\!\cdots\!27}a^{10}+\frac{96\!\cdots\!73}{50\!\cdots\!27}a^{9}-\frac{87\!\cdots\!32}{50\!\cdots\!27}a^{8}-\frac{33\!\cdots\!09}{50\!\cdots\!27}a^{7}+\frac{67\!\cdots\!52}{50\!\cdots\!27}a^{6}+\frac{32\!\cdots\!70}{50\!\cdots\!27}a^{5}-\frac{14\!\cdots\!57}{50\!\cdots\!27}a^{4}-\frac{91\!\cdots\!86}{50\!\cdots\!27}a^{3}+\frac{40\!\cdots\!99}{50\!\cdots\!27}a^{2}+\frac{12\!\cdots\!24}{50\!\cdots\!27}a-\frac{65\!\cdots\!63}{50\!\cdots\!27}$ (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: | \( 105232700409 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 105232700409 \cdot 1}{2\cdot\sqrt{21524562403589040109288671558095801}}\cr\approx \mathstrut & 0.143091222043212 \end{aligned}\] (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
\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.467605011588281.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.16.12.3 | $x^{16} - 84 x^{12} + 6272 x^{8} + 39788 x^{4} + 64827$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |
\(41\) | 41.16.15.1 | $x^{16} + 164$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |