Normalized defining polynomial
\( x^{16} - x^{15} + 15 x^{14} - 31 x^{13} + 124 x^{12} - 279 x^{11} + 693 x^{10} - 1296 x^{9} + 2397 x^{8} + \cdots + 729 \)
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: | \(2555478887462114090409\) \(\medspace = 3^{12}\cdot 37^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.78\) | 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: | $3^{3/4}37^{3/4}\approx 34.197332740531884$ | ||
Ramified primes: | \(3\), \(37\) | 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 not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{2}{9}a^{7}-\frac{2}{9}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{1}{3}a^{7}+\frac{4}{9}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{27}a^{12}-\frac{1}{27}a^{11}+\frac{2}{27}a^{9}-\frac{2}{27}a^{8}-\frac{4}{9}a^{7}+\frac{1}{9}a^{6}+\frac{1}{3}a^{5}+\frac{1}{9}a^{4}$, $\frac{1}{27}a^{13}-\frac{1}{27}a^{11}-\frac{1}{27}a^{10}+\frac{1}{9}a^{9}+\frac{4}{27}a^{8}-\frac{2}{9}a^{7}-\frac{1}{3}a^{6}+\frac{1}{9}a^{5}+\frac{4}{9}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{81}a^{14}-\frac{1}{81}a^{13}+\frac{2}{81}a^{11}-\frac{2}{81}a^{10}-\frac{4}{27}a^{9}+\frac{1}{27}a^{8}+\frac{1}{9}a^{7}-\frac{8}{27}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{47\!\cdots\!19}a^{15}+\frac{23\!\cdots\!96}{47\!\cdots\!19}a^{14}+\frac{3803212873753}{15\!\cdots\!73}a^{13}-\frac{47\!\cdots\!14}{47\!\cdots\!19}a^{12}-\frac{871495038458773}{42\!\cdots\!29}a^{11}-\frac{290068938621071}{15\!\cdots\!73}a^{10}+\frac{494254140000656}{17\!\cdots\!97}a^{9}+\frac{47\!\cdots\!74}{52\!\cdots\!91}a^{8}-\frac{12\!\cdots\!82}{15\!\cdots\!73}a^{7}-\frac{11\!\cdots\!64}{52\!\cdots\!91}a^{6}-\frac{25\!\cdots\!26}{52\!\cdots\!91}a^{5}-\frac{661647551199034}{58\!\cdots\!99}a^{4}-\frac{32\!\cdots\!01}{17\!\cdots\!97}a^{3}+\frac{331066601376889}{19\!\cdots\!33}a^{2}-\frac{749567845308923}{19\!\cdots\!33}a+\frac{542247619313597}{19\!\cdots\!33}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{4}$, which has order $4$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{40115937572}{17807575005891} a^{15} - \frac{13167989095}{17807575005891} a^{14} + \frac{594064940792}{17807575005891} a^{13} - \frac{850985301569}{17807575005891} a^{12} + \frac{402805705061}{1618870455081} a^{11} - \frac{8320256871350}{17807575005891} a^{10} + \frac{7534119932597}{5935858335297} a^{9} - \frac{1398969721465}{659539815033} a^{8} + \frac{24495414544313}{5935858335297} a^{7} - \frac{30184263512378}{5935858335297} a^{6} + \frac{1575204576139}{219846605011} a^{5} - \frac{10868286226390}{1978619445099} a^{4} + \frac{3297898235777}{659539815033} a^{3} + \frac{781540102837}{659539815033} a^{2} - \frac{26162374961}{219846605011} a - \frac{343863337069}{219846605011} \) (order $6$) | 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: | $\frac{574242526082968}{47\!\cdots\!19}a^{15}-\frac{324897276742576}{47\!\cdots\!19}a^{14}+\frac{30\!\cdots\!66}{15\!\cdots\!73}a^{13}-\frac{13\!\cdots\!79}{47\!\cdots\!19}a^{12}+\frac{66\!\cdots\!03}{42\!\cdots\!29}a^{11}-\frac{15\!\cdots\!56}{52\!\cdots\!91}a^{10}+\frac{42\!\cdots\!31}{52\!\cdots\!91}a^{9}-\frac{78\!\cdots\!62}{58\!\cdots\!99}a^{8}+\frac{41\!\cdots\!84}{15\!\cdots\!73}a^{7}-\frac{56\!\cdots\!22}{17\!\cdots\!97}a^{6}+\frac{23\!\cdots\!48}{52\!\cdots\!91}a^{5}-\frac{18\!\cdots\!90}{58\!\cdots\!99}a^{4}+\frac{46\!\cdots\!64}{17\!\cdots\!97}a^{3}+\frac{37\!\cdots\!43}{58\!\cdots\!99}a^{2}+\frac{68007217250317}{19\!\cdots\!33}a-\frac{33\!\cdots\!77}{19\!\cdots\!33}$, $\frac{129139258327309}{47\!\cdots\!19}a^{15}-\frac{91872453178387}{47\!\cdots\!19}a^{14}+\frac{683930166612092}{15\!\cdots\!73}a^{13}-\frac{27\!\cdots\!60}{47\!\cdots\!19}a^{12}+\frac{15\!\cdots\!31}{42\!\cdots\!29}a^{11}-\frac{29\!\cdots\!97}{52\!\cdots\!91}a^{10}+\frac{92\!\cdots\!07}{52\!\cdots\!91}a^{9}-\frac{14\!\cdots\!16}{52\!\cdots\!91}a^{8}+\frac{86\!\cdots\!56}{15\!\cdots\!73}a^{7}-\frac{11\!\cdots\!61}{17\!\cdots\!97}a^{6}+\frac{55\!\cdots\!32}{52\!\cdots\!91}a^{5}-\frac{12\!\cdots\!52}{17\!\cdots\!97}a^{4}+\frac{18\!\cdots\!95}{17\!\cdots\!97}a^{3}+\frac{696985300411421}{58\!\cdots\!99}a^{2}-\frac{230636859343998}{19\!\cdots\!33}a-\frac{37529971241399}{19\!\cdots\!33}$, $\frac{70762075025429}{52\!\cdots\!91}a^{15}-\frac{77666613173657}{15\!\cdots\!73}a^{14}+\frac{31\!\cdots\!82}{15\!\cdots\!73}a^{13}-\frac{15\!\cdots\!22}{52\!\cdots\!91}a^{12}+\frac{21\!\cdots\!27}{14\!\cdots\!43}a^{11}-\frac{44\!\cdots\!88}{15\!\cdots\!73}a^{10}+\frac{40\!\cdots\!59}{52\!\cdots\!91}a^{9}-\frac{67\!\cdots\!62}{52\!\cdots\!91}a^{8}+\frac{43\!\cdots\!36}{17\!\cdots\!97}a^{7}-\frac{16\!\cdots\!35}{52\!\cdots\!91}a^{6}+\frac{73\!\cdots\!77}{17\!\cdots\!97}a^{5}-\frac{18\!\cdots\!86}{58\!\cdots\!99}a^{4}+\frac{52\!\cdots\!63}{19\!\cdots\!33}a^{3}+\frac{13\!\cdots\!05}{19\!\cdots\!33}a^{2}+\frac{182482315860}{19\!\cdots\!33}a-\frac{24\!\cdots\!78}{19\!\cdots\!33}$, $\frac{23609880171170}{52\!\cdots\!91}a^{15}+\frac{43506139479575}{52\!\cdots\!91}a^{14}+\frac{308372408539067}{52\!\cdots\!91}a^{13}+\frac{330967880019253}{52\!\cdots\!91}a^{12}+\frac{115740956503976}{47\!\cdots\!81}a^{11}+\frac{18\!\cdots\!94}{52\!\cdots\!91}a^{10}+\frac{17995326713268}{19\!\cdots\!33}a^{9}+\frac{43\!\cdots\!38}{17\!\cdots\!97}a^{8}-\frac{51\!\cdots\!34}{17\!\cdots\!97}a^{7}+\frac{19\!\cdots\!34}{17\!\cdots\!97}a^{6}-\frac{23\!\cdots\!95}{19\!\cdots\!33}a^{5}+\frac{14\!\cdots\!78}{58\!\cdots\!99}a^{4}-\frac{30\!\cdots\!57}{19\!\cdots\!33}a^{3}+\frac{44\!\cdots\!20}{19\!\cdots\!33}a^{2}+\frac{201359860819500}{19\!\cdots\!33}a-\frac{24\!\cdots\!13}{19\!\cdots\!33}$, $\frac{938712040726475}{47\!\cdots\!19}a^{15}-\frac{902716987150916}{47\!\cdots\!19}a^{14}+\frac{15\!\cdots\!17}{52\!\cdots\!91}a^{13}-\frac{28\!\cdots\!13}{47\!\cdots\!19}a^{12}+\frac{10\!\cdots\!34}{42\!\cdots\!29}a^{11}-\frac{86\!\cdots\!64}{15\!\cdots\!73}a^{10}+\frac{72\!\cdots\!29}{52\!\cdots\!91}a^{9}-\frac{13\!\cdots\!65}{52\!\cdots\!91}a^{8}+\frac{77\!\cdots\!68}{15\!\cdots\!73}a^{7}-\frac{37\!\cdots\!36}{52\!\cdots\!91}a^{6}+\frac{51\!\cdots\!27}{52\!\cdots\!91}a^{5}-\frac{18\!\cdots\!20}{19\!\cdots\!33}a^{4}+\frac{14\!\cdots\!41}{17\!\cdots\!97}a^{3}-\frac{48\!\cdots\!73}{19\!\cdots\!33}a^{2}-\frac{25\!\cdots\!24}{19\!\cdots\!33}a+\frac{480105242851582}{19\!\cdots\!33}$, $\frac{958056611489524}{47\!\cdots\!19}a^{15}+\frac{8149352328056}{47\!\cdots\!19}a^{14}+\frac{46\!\cdots\!28}{15\!\cdots\!73}a^{13}-\frac{15\!\cdots\!35}{47\!\cdots\!19}a^{12}+\frac{86\!\cdots\!62}{42\!\cdots\!29}a^{11}-\frac{52\!\cdots\!77}{15\!\cdots\!73}a^{10}+\frac{48\!\cdots\!59}{52\!\cdots\!91}a^{9}-\frac{73\!\cdots\!23}{52\!\cdots\!91}a^{8}+\frac{42\!\cdots\!58}{15\!\cdots\!73}a^{7}-\frac{13\!\cdots\!02}{52\!\cdots\!91}a^{6}+\frac{19\!\cdots\!27}{52\!\cdots\!91}a^{5}-\frac{15\!\cdots\!53}{19\!\cdots\!33}a^{4}+\frac{72\!\cdots\!38}{17\!\cdots\!97}a^{3}+\frac{29\!\cdots\!97}{58\!\cdots\!99}a^{2}-\frac{23\!\cdots\!68}{19\!\cdots\!33}a-\frac{36\!\cdots\!12}{19\!\cdots\!33}$, $\frac{587374576309}{19\!\cdots\!33}a^{15}+\frac{50190550597351}{15\!\cdots\!73}a^{14}+\frac{660433995798143}{15\!\cdots\!73}a^{13}+\frac{13339059419674}{58\!\cdots\!99}a^{12}+\frac{254554014311593}{14\!\cdots\!43}a^{11}+\frac{24\!\cdots\!93}{15\!\cdots\!73}a^{10}-\frac{12\!\cdots\!02}{52\!\cdots\!91}a^{9}+\frac{38\!\cdots\!43}{17\!\cdots\!97}a^{8}-\frac{90\!\cdots\!14}{17\!\cdots\!97}a^{7}+\frac{73\!\cdots\!67}{52\!\cdots\!91}a^{6}-\frac{40\!\cdots\!35}{17\!\cdots\!97}a^{5}+\frac{69\!\cdots\!64}{17\!\cdots\!97}a^{4}-\frac{25\!\cdots\!46}{58\!\cdots\!99}a^{3}+\frac{22\!\cdots\!22}{58\!\cdots\!99}a^{2}-\frac{37\!\cdots\!22}{19\!\cdots\!33}a+\frac{291834774578233}{19\!\cdots\!33}$ | 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: | \( 21312.5774141 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 21312.5774141 \cdot 4}{6\cdot\sqrt{2555478887462114090409}}\cr\approx \mathstrut & 0.682727683259 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 16T42):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4\wr C_2$ |
Character table for $C_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{37}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-111}) \), 4.2.4107.1 x2, 4.0.333.1 x2, \(\Q(\sqrt{-3}, \sqrt{37})\), 8.0.36926037.1, 8.0.50551744653.2, 8.0.151807041.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 8 siblings: | 8.0.50551744653.2, 8.0.36926037.1 |
Degree 16 sibling: | 16.4.3498450596935634189769921.3 |
Minimal sibling: | 8.0.36926037.1 |
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 | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(37\) | 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
37.4.3.1 | $x^{4} + 111$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
37.4.3.1 | $x^{4} + 111$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |