Normalized defining polynomial
\( x^{16} - 7 x^{15} + 96 x^{14} + 91 x^{13} + 2772 x^{12} + 129667 x^{11} - 310036 x^{10} + \cdots + 106045326871552 \)
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)
| |
Discriminant: | \(462732306245995722656474121747020143758680081\) \(\medspace = 29^{14}\cdot 41^{15}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(618.85\) | 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: | $29^{7/8}41^{15/16}\approx 618.8467490579351$ | ||
Ramified primes: | \(29\), \(41\) | 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 a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{10}-\frac{1}{32}a^{9}+\frac{1}{32}a^{8}-\frac{1}{16}a^{7}+\frac{3}{32}a^{6}-\frac{5}{32}a^{5}-\frac{5}{32}a^{4}+\frac{1}{4}a$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{8}+\frac{1}{32}a^{7}-\frac{1}{16}a^{6}+\frac{3}{16}a^{5}-\frac{5}{32}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{12}-\frac{1}{64}a^{9}+\frac{1}{64}a^{8}-\frac{1}{32}a^{7}+\frac{3}{32}a^{6}+\frac{11}{64}a^{5}-\frac{1}{8}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{13}-\frac{1}{128}a^{11}-\frac{3}{256}a^{10}+\frac{3}{256}a^{9}-\frac{1}{128}a^{8}-\frac{3}{32}a^{7}+\frac{9}{256}a^{6}-\frac{1}{128}a^{5}+\frac{11}{64}a^{4}+\frac{3}{32}a^{3}+\frac{7}{16}a^{2}-\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{42496}a^{14}+\frac{1}{2656}a^{13}+\frac{31}{21248}a^{12}+\frac{397}{42496}a^{11}+\frac{259}{42496}a^{10}-\frac{33}{21248}a^{9}-\frac{167}{5312}a^{8}-\frac{3815}{42496}a^{7}-\frac{1953}{21248}a^{6}-\frac{2129}{10624}a^{5}+\frac{631}{5312}a^{4}-\frac{1217}{2656}a^{3}-\frac{57}{1328}a^{2}+\frac{11}{332}a+\frac{15}{83}$, $\frac{1}{18\!\cdots\!64}a^{15}+\frac{75\!\cdots\!61}{11\!\cdots\!08}a^{14}-\frac{16\!\cdots\!07}{23\!\cdots\!08}a^{13}+\frac{97\!\cdots\!53}{18\!\cdots\!64}a^{12}-\frac{24\!\cdots\!75}{18\!\cdots\!64}a^{11}+\frac{20\!\cdots\!79}{92\!\cdots\!32}a^{10}-\frac{36\!\cdots\!41}{66\!\cdots\!88}a^{9}-\frac{17\!\cdots\!83}{18\!\cdots\!64}a^{8}+\frac{23\!\cdots\!83}{23\!\cdots\!08}a^{7}+\frac{75\!\cdots\!61}{92\!\cdots\!32}a^{6}-\frac{71\!\cdots\!85}{46\!\cdots\!16}a^{5}-\frac{35\!\cdots\!27}{23\!\cdots\!08}a^{4}-\frac{22\!\cdots\!97}{11\!\cdots\!04}a^{3}+\frac{35\!\cdots\!09}{57\!\cdots\!52}a^{2}+\frac{23\!\cdots\!69}{28\!\cdots\!76}a-\frac{22\!\cdots\!11}{72\!\cdots\!94}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{13506148}$, which has order $13506148$ (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: | $\frac{28\!\cdots\!85}{62\!\cdots\!28}a^{15}-\frac{92\!\cdots\!65}{10\!\cdots\!92}a^{14}+\frac{30\!\cdots\!35}{31\!\cdots\!64}a^{13}-\frac{75\!\cdots\!65}{62\!\cdots\!28}a^{12}+\frac{91\!\cdots\!35}{15\!\cdots\!32}a^{11}-\frac{35\!\cdots\!35}{62\!\cdots\!28}a^{10}-\frac{16\!\cdots\!65}{22\!\cdots\!76}a^{9}+\frac{33\!\cdots\!25}{62\!\cdots\!28}a^{8}-\frac{17\!\cdots\!85}{62\!\cdots\!28}a^{7}+\frac{93\!\cdots\!15}{31\!\cdots\!64}a^{6}-\frac{11\!\cdots\!05}{19\!\cdots\!79}a^{5}-\frac{34\!\cdots\!05}{15\!\cdots\!32}a^{4}+\frac{10\!\cdots\!75}{19\!\cdots\!79}a^{3}+\frac{22\!\cdots\!65}{38\!\cdots\!58}a^{2}+\frac{34\!\cdots\!00}{19\!\cdots\!79}a+\frac{58\!\cdots\!83}{19\!\cdots\!79}$, $\frac{46\!\cdots\!35}{25\!\cdots\!64}a^{15}+\frac{44\!\cdots\!99}{21\!\cdots\!48}a^{14}-\frac{67\!\cdots\!15}{15\!\cdots\!04}a^{13}+\frac{17\!\cdots\!87}{25\!\cdots\!64}a^{12}-\frac{16\!\cdots\!57}{25\!\cdots\!64}a^{11}+\frac{65\!\cdots\!37}{12\!\cdots\!32}a^{10}+\frac{45\!\cdots\!75}{91\!\cdots\!88}a^{9}-\frac{74\!\cdots\!13}{25\!\cdots\!64}a^{8}+\frac{29\!\cdots\!61}{79\!\cdots\!02}a^{7}-\frac{36\!\cdots\!75}{12\!\cdots\!32}a^{6}+\frac{36\!\cdots\!81}{31\!\cdots\!08}a^{5}+\frac{26\!\cdots\!27}{79\!\cdots\!02}a^{4}-\frac{26\!\cdots\!08}{39\!\cdots\!01}a^{3}+\frac{16\!\cdots\!84}{39\!\cdots\!01}a^{2}-\frac{74\!\cdots\!56}{39\!\cdots\!01}a+\frac{20\!\cdots\!91}{39\!\cdots\!01}$, $\frac{66\!\cdots\!11}{50\!\cdots\!28}a^{15}-\frac{78\!\cdots\!09}{86\!\cdots\!92}a^{14}+\frac{21\!\cdots\!23}{25\!\cdots\!64}a^{13}-\frac{58\!\cdots\!91}{50\!\cdots\!28}a^{12}+\frac{97\!\cdots\!83}{25\!\cdots\!64}a^{11}-\frac{22\!\cdots\!91}{50\!\cdots\!28}a^{10}-\frac{16\!\cdots\!99}{18\!\cdots\!76}a^{9}+\frac{13\!\cdots\!27}{50\!\cdots\!28}a^{8}-\frac{17\!\cdots\!25}{50\!\cdots\!28}a^{7}+\frac{57\!\cdots\!95}{25\!\cdots\!64}a^{6}+\frac{16\!\cdots\!05}{31\!\cdots\!08}a^{5}-\frac{90\!\cdots\!53}{63\!\cdots\!16}a^{4}+\frac{10\!\cdots\!39}{31\!\cdots\!08}a^{3}+\frac{33\!\cdots\!77}{79\!\cdots\!02}a^{2}+\frac{19\!\cdots\!46}{39\!\cdots\!01}a+\frac{77\!\cdots\!05}{39\!\cdots\!01}$, $\frac{11\!\cdots\!33}{15\!\cdots\!12}a^{15}-\frac{33\!\cdots\!99}{19\!\cdots\!28}a^{14}+\frac{29\!\cdots\!75}{18\!\cdots\!64}a^{13}-\frac{22\!\cdots\!81}{15\!\cdots\!12}a^{12}-\frac{18\!\cdots\!27}{18\!\cdots\!64}a^{11}+\frac{20\!\cdots\!19}{15\!\cdots\!12}a^{10}-\frac{33\!\cdots\!67}{27\!\cdots\!52}a^{9}-\frac{24\!\cdots\!79}{15\!\cdots\!12}a^{8}+\frac{13\!\cdots\!23}{15\!\cdots\!12}a^{7}-\frac{15\!\cdots\!65}{37\!\cdots\!28}a^{6}-\frac{12\!\cdots\!49}{94\!\cdots\!32}a^{5}+\frac{12\!\cdots\!93}{94\!\cdots\!32}a^{4}-\frac{50\!\cdots\!97}{11\!\cdots\!54}a^{3}+\frac{22\!\cdots\!55}{23\!\cdots\!08}a^{2}+\frac{33\!\cdots\!79}{23\!\cdots\!08}a-\frac{65\!\cdots\!34}{59\!\cdots\!27}$, $\frac{62\!\cdots\!49}{94\!\cdots\!32}a^{15}-\frac{35\!\cdots\!53}{49\!\cdots\!32}a^{14}+\frac{23\!\cdots\!05}{18\!\cdots\!64}a^{13}-\frac{32\!\cdots\!47}{18\!\cdots\!64}a^{12}+\frac{53\!\cdots\!03}{37\!\cdots\!28}a^{11}-\frac{40\!\cdots\!93}{37\!\cdots\!28}a^{10}-\frac{41\!\cdots\!05}{34\!\cdots\!44}a^{9}+\frac{66\!\cdots\!13}{94\!\cdots\!32}a^{8}-\frac{29\!\cdots\!89}{37\!\cdots\!28}a^{7}+\frac{30\!\cdots\!85}{47\!\cdots\!16}a^{6}-\frac{38\!\cdots\!93}{11\!\cdots\!54}a^{5}-\frac{11\!\cdots\!21}{23\!\cdots\!08}a^{4}+\frac{96\!\cdots\!77}{59\!\cdots\!27}a^{3}-\frac{63\!\cdots\!36}{59\!\cdots\!27}a^{2}+\frac{29\!\cdots\!39}{59\!\cdots\!27}a-\frac{10\!\cdots\!29}{59\!\cdots\!27}$, $\frac{40\!\cdots\!45}{75\!\cdots\!56}a^{15}-\frac{72\!\cdots\!21}{98\!\cdots\!64}a^{14}+\frac{38\!\cdots\!11}{37\!\cdots\!28}a^{13}-\frac{47\!\cdots\!97}{75\!\cdots\!56}a^{12}+\frac{21\!\cdots\!37}{37\!\cdots\!28}a^{11}+\frac{26\!\cdots\!43}{75\!\cdots\!56}a^{10}-\frac{11\!\cdots\!67}{27\!\cdots\!52}a^{9}+\frac{24\!\cdots\!77}{75\!\cdots\!56}a^{8}-\frac{19\!\cdots\!67}{75\!\cdots\!56}a^{7}+\frac{35\!\cdots\!31}{37\!\cdots\!28}a^{6}+\frac{51\!\cdots\!69}{47\!\cdots\!16}a^{5}-\frac{67\!\cdots\!21}{94\!\cdots\!32}a^{4}-\frac{21\!\cdots\!69}{47\!\cdots\!16}a^{3}+\frac{15\!\cdots\!93}{11\!\cdots\!54}a^{2}-\frac{55\!\cdots\!05}{59\!\cdots\!27}a+\frac{50\!\cdots\!55}{59\!\cdots\!27}$, $\frac{27\!\cdots\!09}{15\!\cdots\!12}a^{15}-\frac{28\!\cdots\!79}{19\!\cdots\!28}a^{14}+\frac{55\!\cdots\!17}{94\!\cdots\!32}a^{13}+\frac{26\!\cdots\!07}{15\!\cdots\!12}a^{12}+\frac{42\!\cdots\!49}{18\!\cdots\!64}a^{11}+\frac{90\!\cdots\!67}{15\!\cdots\!12}a^{10}-\frac{19\!\cdots\!05}{27\!\cdots\!52}a^{9}-\frac{65\!\cdots\!91}{15\!\cdots\!12}a^{8}+\frac{10\!\cdots\!19}{15\!\cdots\!12}a^{7}+\frac{24\!\cdots\!61}{37\!\cdots\!28}a^{6}-\frac{15\!\cdots\!01}{94\!\cdots\!32}a^{5}+\frac{16\!\cdots\!31}{94\!\cdots\!32}a^{4}-\frac{38\!\cdots\!55}{23\!\cdots\!08}a^{3}+\frac{13\!\cdots\!63}{23\!\cdots\!08}a^{2}+\frac{47\!\cdots\!11}{23\!\cdots\!08}a+\frac{66\!\cdots\!02}{59\!\cdots\!27}$ (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: | \( 276024556298 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 276024556298 \cdot 13506148}{2\cdot\sqrt{462732306245995722656474121747020143758680081}}\cr\approx \mathstrut & 210.486024391043 \end{aligned}\] (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
\(\Q(\sqrt{41}) \), 4.4.57962561.1, 8.8.115844383968839978801.2 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{8}$ | $16$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | $16$ | $16$ | ${\href{/padicField/59.1.0.1}{1} }^{16}$ |
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 |
---|---|---|---|---|---|---|---|
\(29\) | 29.16.14.5 | $x^{16} - 696 x^{8} + 1682$ | $8$ | $2$ | $14$ | $C_{16} : C_2$ | $[\ ]_{8}^{4}$ |
\(41\) | 41.16.15.4 | $x^{16} + 82$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |