Normalized defining polynomial
\( x^{16} - x^{15} + 46 x^{14} - 105 x^{13} + 1270 x^{12} - 4449 x^{11} + 16060 x^{10} - 63735 x^{9} + 211046 x^{8} - 1759715 x^{7} + 8594450 x^{6} - 26273899 x^{5} + \cdots + 155285056 \)
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: | \(9260065233133681348183837890625\) \(\medspace = 5^{12}\cdot 41^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(86.18\) | 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: | $5^{3/4}41^{7/8}\approx 86.18136686908747$ | ||
Ramified primes: | \(5\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{8}a^{8}+\frac{3}{16}a^{5}-\frac{3}{16}a^{4}+\frac{5}{16}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{9}-\frac{1}{4}a^{7}+\frac{3}{16}a^{6}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{64}a^{13}-\frac{1}{32}a^{12}+\frac{1}{64}a^{11}-\frac{1}{64}a^{9}-\frac{1}{16}a^{8}+\frac{15}{64}a^{7}-\frac{3}{32}a^{6}+\frac{11}{64}a^{5}-\frac{11}{64}a^{3}+\frac{5}{16}a^{2}-\frac{5}{16}a-\frac{1}{4}$, $\frac{1}{128}a^{14}-\frac{3}{128}a^{12}-\frac{1}{64}a^{11}-\frac{5}{128}a^{10}+\frac{3}{64}a^{9}-\frac{9}{128}a^{8}+\frac{1}{16}a^{7}-\frac{1}{128}a^{6}-\frac{11}{64}a^{5}+\frac{9}{128}a^{4}+\frac{1}{64}a^{3}+\frac{9}{32}a^{2}+\frac{3}{16}a+\frac{1}{4}$, $\frac{1}{10\!\cdots\!64}a^{15}-\frac{54\!\cdots\!59}{25\!\cdots\!16}a^{14}-\frac{62\!\cdots\!03}{10\!\cdots\!64}a^{13}+\frac{13\!\cdots\!41}{51\!\cdots\!32}a^{12}-\frac{24\!\cdots\!41}{10\!\cdots\!64}a^{11}-\frac{10\!\cdots\!67}{51\!\cdots\!32}a^{10}+\frac{78\!\cdots\!03}{10\!\cdots\!64}a^{9}+\frac{15\!\cdots\!85}{25\!\cdots\!16}a^{8}-\frac{29\!\cdots\!33}{10\!\cdots\!64}a^{7}-\frac{42\!\cdots\!69}{51\!\cdots\!32}a^{6}+\frac{78\!\cdots\!81}{10\!\cdots\!64}a^{5}-\frac{46\!\cdots\!77}{51\!\cdots\!32}a^{4}-\frac{94\!\cdots\!35}{12\!\cdots\!08}a^{3}-\frac{17\!\cdots\!71}{12\!\cdots\!08}a^{2}-\frac{25\!\cdots\!73}{64\!\cdots\!04}a-\frac{13\!\cdots\!01}{16\!\cdots\!26}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{300}$, which has order $600$ (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: | $\frac{42\!\cdots\!86}{17\!\cdots\!33}a^{15}-\frac{13\!\cdots\!19}{56\!\cdots\!56}a^{14}+\frac{51\!\cdots\!71}{56\!\cdots\!56}a^{13}-\frac{15\!\cdots\!57}{56\!\cdots\!56}a^{12}+\frac{13\!\cdots\!01}{56\!\cdots\!56}a^{11}-\frac{57\!\cdots\!01}{56\!\cdots\!56}a^{10}+\frac{12\!\cdots\!51}{56\!\cdots\!56}a^{9}-\frac{59\!\cdots\!01}{56\!\cdots\!56}a^{8}+\frac{26\!\cdots\!33}{56\!\cdots\!56}a^{7}-\frac{19\!\cdots\!87}{56\!\cdots\!56}a^{6}+\frac{10\!\cdots\!63}{56\!\cdots\!56}a^{5}-\frac{23\!\cdots\!99}{56\!\cdots\!56}a^{4}+\frac{31\!\cdots\!81}{56\!\cdots\!56}a^{3}+\frac{85\!\cdots\!53}{35\!\cdots\!66}a^{2}+\frac{12\!\cdots\!47}{14\!\cdots\!64}a-\frac{35\!\cdots\!51}{35\!\cdots\!66}$, $\frac{16\!\cdots\!21}{22\!\cdots\!24}a^{15}+\frac{78\!\cdots\!09}{11\!\cdots\!12}a^{14}+\frac{76\!\cdots\!87}{22\!\cdots\!24}a^{13}-\frac{39\!\cdots\!39}{56\!\cdots\!56}a^{12}+\frac{20\!\cdots\!65}{22\!\cdots\!24}a^{11}-\frac{76\!\cdots\!49}{56\!\cdots\!56}a^{10}+\frac{19\!\cdots\!25}{22\!\cdots\!24}a^{9}-\frac{29\!\cdots\!37}{11\!\cdots\!12}a^{8}+\frac{22\!\cdots\!37}{22\!\cdots\!24}a^{7}-\frac{59\!\cdots\!41}{56\!\cdots\!56}a^{6}+\frac{89\!\cdots\!83}{22\!\cdots\!24}a^{5}-\frac{60\!\cdots\!19}{56\!\cdots\!56}a^{4}+\frac{66\!\cdots\!23}{11\!\cdots\!12}a^{3}+\frac{69\!\cdots\!77}{28\!\cdots\!28}a^{2}+\frac{17\!\cdots\!47}{28\!\cdots\!28}a-\frac{32\!\cdots\!33}{70\!\cdots\!32}$, $\frac{29\!\cdots\!71}{11\!\cdots\!12}a^{15}+\frac{48\!\cdots\!05}{11\!\cdots\!12}a^{14}+\frac{73\!\cdots\!97}{56\!\cdots\!56}a^{13}+\frac{81\!\cdots\!05}{11\!\cdots\!12}a^{12}+\frac{19\!\cdots\!01}{56\!\cdots\!56}a^{11}-\frac{29\!\cdots\!95}{11\!\cdots\!12}a^{10}+\frac{89\!\cdots\!43}{28\!\cdots\!28}a^{9}-\frac{92\!\cdots\!89}{11\!\cdots\!12}a^{8}+\frac{16\!\cdots\!85}{56\!\cdots\!56}a^{7}-\frac{42\!\cdots\!33}{11\!\cdots\!12}a^{6}+\frac{68\!\cdots\!31}{56\!\cdots\!56}a^{5}-\frac{37\!\cdots\!57}{11\!\cdots\!12}a^{4}+\frac{10\!\cdots\!77}{11\!\cdots\!12}a^{3}+\frac{15\!\cdots\!53}{14\!\cdots\!64}a^{2}+\frac{12\!\cdots\!83}{28\!\cdots\!28}a-\frac{15\!\cdots\!69}{70\!\cdots\!32}$, $\frac{12\!\cdots\!87}{22\!\cdots\!24}a^{15}+\frac{12\!\cdots\!41}{11\!\cdots\!12}a^{14}+\frac{70\!\cdots\!03}{22\!\cdots\!24}a^{13}+\frac{15\!\cdots\!49}{56\!\cdots\!56}a^{12}+\frac{18\!\cdots\!45}{22\!\cdots\!24}a^{11}-\frac{11\!\cdots\!43}{28\!\cdots\!28}a^{10}+\frac{19\!\cdots\!01}{22\!\cdots\!24}a^{9}-\frac{21\!\cdots\!17}{11\!\cdots\!12}a^{8}+\frac{12\!\cdots\!37}{22\!\cdots\!24}a^{7}-\frac{48\!\cdots\!57}{56\!\cdots\!56}a^{6}+\frac{55\!\cdots\!59}{22\!\cdots\!24}a^{5}-\frac{24\!\cdots\!37}{28\!\cdots\!28}a^{4}+\frac{26\!\cdots\!47}{28\!\cdots\!28}a^{3}-\frac{42\!\cdots\!15}{35\!\cdots\!66}a^{2}-\frac{15\!\cdots\!23}{14\!\cdots\!64}a+\frac{25\!\cdots\!23}{35\!\cdots\!66}$, $\frac{44\!\cdots\!53}{11\!\cdots\!12}a^{15}+\frac{43\!\cdots\!05}{14\!\cdots\!64}a^{14}+\frac{76\!\cdots\!27}{14\!\cdots\!64}a^{13}+\frac{45\!\cdots\!25}{28\!\cdots\!28}a^{12}-\frac{21\!\cdots\!37}{56\!\cdots\!56}a^{11}+\frac{25\!\cdots\!03}{56\!\cdots\!56}a^{10}-\frac{16\!\cdots\!52}{17\!\cdots\!33}a^{9}+\frac{70\!\cdots\!17}{28\!\cdots\!28}a^{8}+\frac{67\!\cdots\!21}{28\!\cdots\!28}a^{7}+\frac{70\!\cdots\!59}{28\!\cdots\!28}a^{6}-\frac{19\!\cdots\!19}{56\!\cdots\!56}a^{5}+\frac{91\!\cdots\!25}{56\!\cdots\!56}a^{4}-\frac{35\!\cdots\!69}{11\!\cdots\!12}a^{3}+\frac{14\!\cdots\!69}{28\!\cdots\!28}a^{2}+\frac{17\!\cdots\!29}{28\!\cdots\!28}a-\frac{55\!\cdots\!39}{70\!\cdots\!32}$, $\frac{78\!\cdots\!49}{28\!\cdots\!28}a^{15}-\frac{15\!\cdots\!95}{11\!\cdots\!12}a^{14}+\frac{14\!\cdots\!17}{11\!\cdots\!12}a^{13}-\frac{24\!\cdots\!41}{11\!\cdots\!12}a^{12}+\frac{39\!\cdots\!79}{11\!\cdots\!12}a^{11}-\frac{11\!\cdots\!21}{11\!\cdots\!12}a^{10}+\frac{47\!\cdots\!65}{11\!\cdots\!12}a^{9}-\frac{17\!\cdots\!65}{11\!\cdots\!12}a^{8}+\frac{59\!\cdots\!39}{11\!\cdots\!12}a^{7}-\frac{51\!\cdots\!55}{11\!\cdots\!12}a^{6}+\frac{23\!\cdots\!13}{11\!\cdots\!12}a^{5}-\frac{74\!\cdots\!95}{11\!\cdots\!12}a^{4}+\frac{10\!\cdots\!23}{11\!\cdots\!12}a^{3}-\frac{22\!\cdots\!83}{14\!\cdots\!64}a^{2}-\frac{41\!\cdots\!43}{28\!\cdots\!28}a+\frac{74\!\cdots\!29}{70\!\cdots\!32}$, $\frac{15\!\cdots\!79}{11\!\cdots\!12}a^{15}-\frac{28\!\cdots\!65}{11\!\cdots\!12}a^{14}+\frac{42\!\cdots\!55}{28\!\cdots\!28}a^{13}-\frac{14\!\cdots\!85}{11\!\cdots\!12}a^{12}+\frac{32\!\cdots\!61}{35\!\cdots\!66}a^{11}-\frac{39\!\cdots\!97}{11\!\cdots\!12}a^{10}+\frac{24\!\cdots\!95}{56\!\cdots\!56}a^{9}-\frac{33\!\cdots\!91}{11\!\cdots\!12}a^{8}+\frac{76\!\cdots\!17}{70\!\cdots\!32}a^{7}-\frac{48\!\cdots\!31}{11\!\cdots\!12}a^{6}+\frac{15\!\cdots\!59}{35\!\cdots\!66}a^{5}-\frac{14\!\cdots\!91}{11\!\cdots\!12}a^{4}+\frac{29\!\cdots\!75}{11\!\cdots\!12}a^{3}-\frac{74\!\cdots\!47}{14\!\cdots\!64}a^{2}-\frac{13\!\cdots\!99}{28\!\cdots\!28}a-\frac{94\!\cdots\!23}{70\!\cdots\!32}$ (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: | \( 10205497.328 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10205497.328 \cdot 600}{2\cdot\sqrt{9260065233133681348183837890625}}\cr\approx \mathstrut & 2.4439220618 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2:C_8$ (as 16T24):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_2^2 : C_8$ |
Character table for $C_2^2 : C_8$ |
Intermediate fields
\(\Q(\sqrt{41}) \), 4.0.8405.1, 4.4.1723025.1, 4.0.344605.1, 8.8.3043035529390625.1, 8.0.3043035529390625.1, 8.0.2968815150625.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 sibling: | 16.0.9260065233133681348183837890625.6 |
Minimal sibling: | 16.0.9260065233133681348183837890625.6 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.4 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.4 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(41\) | 41.16.14.1 | $x^{16} + 304 x^{15} + 40480 x^{14} + 3085600 x^{13} + 147416080 x^{12} + 4529584192 x^{11} + 87831092608 x^{10} + 996302227840 x^{9} + 5391168776882 x^{8} + 5977813379504 x^{7} + 3161920977824 x^{6} + 978514601120 x^{5} + 196936323920 x^{4} + 202153692608 x^{3} + 3372805705856 x^{2} + 36445904670848 x + 172395305267889$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |