Normalized defining polynomial
\( x^{16} - 2 x^{15} - 6 x^{14} - 11 x^{13} + 114 x^{12} - 15 x^{11} + 62 x^{10} + 193 x^{9} + 828 x^{8} + \cdots + 1 \)
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: | \(35847274805742431640625\) \(\medspace = 5^{12}\cdot 59^{8}\) | 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: | \(25.68\) | 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: | $5^{3/4}59^{1/2}\approx 25.683458749990002$ | ||
Ramified primes: | \(5\), \(59\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{5}+\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{15}a^{10}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{15}$, $\frac{1}{75}a^{11}-\frac{2}{75}a^{10}-\frac{2}{25}a^{9}+\frac{4}{75}a^{8}-\frac{26}{75}a^{7}-\frac{11}{75}a^{6}+\frac{3}{25}a^{5}-\frac{31}{75}a^{4}+\frac{19}{75}a^{3}-\frac{2}{25}a^{2}-\frac{17}{75}a+\frac{26}{75}$, $\frac{1}{75}a^{12}+\frac{7}{75}a^{9}-\frac{1}{25}a^{8}-\frac{23}{75}a^{7}+\frac{2}{75}a^{6}+\frac{9}{25}a^{5}+\frac{22}{75}a^{4}-\frac{28}{75}a^{3}+\frac{11}{75}a^{2}-\frac{6}{25}a-\frac{6}{25}$, $\frac{1}{75}a^{13}+\frac{2}{75}a^{10}-\frac{1}{25}a^{9}+\frac{7}{75}a^{8}+\frac{9}{25}a^{7}-\frac{6}{25}a^{6}-\frac{28}{75}a^{5}-\frac{23}{75}a^{4}+\frac{11}{75}a^{3}+\frac{37}{75}a^{2}+\frac{32}{75}a+\frac{7}{15}$, $\frac{1}{55875}a^{14}+\frac{1}{55875}a^{13}+\frac{371}{55875}a^{12}-\frac{14}{55875}a^{11}+\frac{1001}{55875}a^{10}-\frac{1246}{18625}a^{9}-\frac{1851}{18625}a^{8}-\frac{7918}{55875}a^{7}-\frac{26413}{55875}a^{6}+\frac{27637}{55875}a^{5}-\frac{25414}{55875}a^{4}-\frac{4443}{18625}a^{3}+\frac{20411}{55875}a^{2}+\frac{19376}{55875}a-\frac{16639}{55875}$, $\frac{1}{2455926453375}a^{15}+\frac{9858662}{2455926453375}a^{14}-\frac{20461334}{5494242625}a^{13}-\frac{5273693626}{818642151125}a^{12}+\frac{3165131734}{818642151125}a^{11}-\frac{20074926674}{818642151125}a^{10}-\frac{342160317}{818642151125}a^{9}-\frac{94528080931}{2455926453375}a^{8}-\frac{1183849693441}{2455926453375}a^{7}-\frac{386989860212}{818642151125}a^{6}-\frac{587042204012}{2455926453375}a^{5}+\frac{832007882137}{2455926453375}a^{4}+\frac{63242201192}{2455926453375}a^{3}+\frac{31907342912}{2455926453375}a^{2}-\frac{165052393496}{818642151125}a-\frac{969769593539}{2455926453375}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{6}$, which has order $6$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{8903471516}{818642151125}a^{15}-\frac{24078588593}{818642151125}a^{14}-\frac{37069123938}{818642151125}a^{13}-\frac{69513900323}{818642151125}a^{12}+\frac{1064453500562}{818642151125}a^{11}-\frac{881003982087}{818642151125}a^{10}+\frac{1095783244474}{818642151125}a^{9}+\frac{1089479389944}{818642151125}a^{8}+\frac{6443202372034}{818642151125}a^{7}-\frac{1873602161411}{818642151125}a^{6}+\frac{9370969606298}{818642151125}a^{5}+\frac{9061266320842}{818642151125}a^{4}+\frac{799450957082}{818642151125}a^{3}+\frac{456941328807}{818642151125}a^{2}-\frac{87387675823}{818642151125}a-\frac{495065252849}{818642151125}$, $\frac{219777320639}{2455926453375}a^{15}-\frac{404157899509}{2455926453375}a^{14}-\frac{463101583363}{818642151125}a^{13}-\frac{2627831814839}{2455926453375}a^{12}+\frac{24661657994396}{2455926453375}a^{11}+\frac{143900415826}{491185290675}a^{10}+\frac{13066710407507}{2455926453375}a^{9}+\frac{44856375034117}{2455926453375}a^{8}+\frac{189043997970157}{2455926453375}a^{7}+\frac{95166401946217}{2455926453375}a^{6}+\frac{220040311118528}{2455926453375}a^{5}+\frac{137563671636372}{818642151125}a^{4}+\frac{266672483269306}{2455926453375}a^{3}+\frac{61640531823871}{2455926453375}a^{2}+\frac{4587139070307}{818642151125}a-\frac{251559598746}{818642151125}$, $\frac{3014546122}{163728430225}a^{15}-\frac{53635818523}{818642151125}a^{14}-\frac{115380186529}{2455926453375}a^{13}-\frac{33422894158}{818642151125}a^{12}+\frac{5827221742966}{2455926453375}a^{11}-\frac{2963741485503}{818642151125}a^{10}+\frac{1847916552809}{818642151125}a^{9}+\frac{1550056750229}{818642151125}a^{8}+\frac{24177750655397}{2455926453375}a^{7}-\frac{41348404615148}{2455926453375}a^{6}+\frac{35062047842297}{2455926453375}a^{5}+\frac{5852938895257}{818642151125}a^{4}-\frac{66555902736484}{2455926453375}a^{3}-\frac{9723772445718}{818642151125}a^{2}+\frac{9965267611036}{2455926453375}a-\frac{1288241676289}{2455926453375}$, $\frac{38500913351}{2455926453375}a^{15}-\frac{12379664286}{818642151125}a^{14}-\frac{313694242588}{2455926453375}a^{13}-\frac{646861181183}{2455926453375}a^{12}+\frac{3940240681982}{2455926453375}a^{11}+\frac{1320579158881}{818642151125}a^{10}+\frac{459751086858}{818642151125}a^{9}+\frac{11140128872269}{2455926453375}a^{8}+\frac{38251832581009}{2455926453375}a^{7}+\frac{45291167338339}{2455926453375}a^{6}+\frac{15710745235446}{818642151125}a^{5}+\frac{109722462662782}{2455926453375}a^{4}+\frac{96771007664482}{2455926453375}a^{3}+\frac{45707513353342}{2455926453375}a^{2}+\frac{12015980270432}{2455926453375}a+\frac{958633458871}{2455926453375}$, $\frac{34683578146}{491185290675}a^{15}-\frac{250606669678}{2455926453375}a^{14}-\frac{424108914236}{818642151125}a^{13}-\frac{803816029526}{818642151125}a^{12}+\frac{18964454800157}{2455926453375}a^{11}+\frac{8816285453312}{2455926453375}a^{10}+\frac{4842738152509}{2455926453375}a^{9}+\frac{39382034194879}{2455926453375}a^{8}+\frac{53539103503383}{818642151125}a^{7}+\frac{122984586144734}{2455926453375}a^{6}+\frac{1076838850621}{16482727875}a^{5}+\frac{374115060700637}{2455926453375}a^{4}+\frac{291212645775062}{2455926453375}a^{3}+\frac{36122487293027}{2455926453375}a^{2}-\frac{7497072186316}{818642151125}a+\frac{934231762549}{818642151125}$, $\frac{78040388431}{491185290675}a^{15}-\frac{162382335028}{491185290675}a^{14}-\frac{1023178458}{1098848525}a^{13}-\frac{818321268926}{491185290675}a^{12}+\frac{119726624433}{6549137209}a^{11}-\frac{622017438408}{163728430225}a^{10}+\frac{4745559050209}{491185290675}a^{9}+\frac{14560299562109}{491185290675}a^{8}+\frac{21143028513403}{163728430225}a^{7}+\frac{17526848954819}{491185290675}a^{6}+\frac{70930930518853}{491185290675}a^{5}+\frac{126483702593702}{491185290675}a^{4}+\frac{11987043856973}{98237058135}a^{3}-\frac{238853625388}{98237058135}a^{2}-\frac{213421477962}{163728430225}a-\frac{1228949358257}{491185290675}$, $\frac{73655447826}{818642151125}a^{15}-\frac{429386335009}{2455926453375}a^{14}-\frac{1328013480769}{2455926453375}a^{13}-\frac{2564930735794}{2455926453375}a^{12}+\frac{8314935781877}{818642151125}a^{11}-\frac{2061184062011}{2455926453375}a^{10}+\frac{16236393708022}{2455926453375}a^{9}+\frac{41537717970382}{2455926453375}a^{8}+\frac{187377434656427}{2455926453375}a^{7}+\frac{80016510364342}{2455926453375}a^{6}+\frac{229996901252989}{2455926453375}a^{5}+\frac{388404439093481}{2455926453375}a^{4}+\frac{81926018970942}{818642151125}a^{3}+\frac{65791436993546}{2455926453375}a^{2}+\frac{2360933581837}{818642151125}a-\frac{466544176322}{2455926453375}$ (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: | \( 11967.7452041 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11967.7452041 \cdot 6}{2\cdot\sqrt{35847274805742431640625}}\cr\approx \mathstrut & 0.460621773783 \end{aligned}\] (assuming GRH)
Galois group
$\SD_{16}$ (as 16T12):
A solvable group of order 16 |
The 7 conjugacy class representatives for $QD_{16}$ |
Character table for $QD_{16}$ |
Intermediate fields
\(\Q(\sqrt{-295}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{5}, \sqrt{-59})\), 4.2.1475.1 x2, 4.0.17405.1 x2, 8.0.7573350625.1, 8.2.3209046875.2 x4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | 8.2.3209046875.2 |
Minimal sibling: | 8.2.3209046875.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | R |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(59\) | 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |