Normalized defining polynomial
\( x^{16} - 4 x^{15} + 7 x^{14} - 7 x^{13} - 10 x^{12} + 7 x^{11} + 182 x^{10} - 411 x^{9} + 346 x^{8} + \cdots + 20 \)
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: | \(19155446635260009765625\) \(\medspace = 5^{14}\cdot 11^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.70\) | 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: | $5^{7/8}11^{3/4}\approx 24.69694688202418$ | ||
Ramified primes: | \(5\), \(11\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{3}{8}a^{5}+\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{49\!\cdots\!72}a^{15}+\frac{11\!\cdots\!81}{49\!\cdots\!72}a^{14}-\frac{62\!\cdots\!75}{49\!\cdots\!72}a^{13}+\frac{80\!\cdots\!09}{49\!\cdots\!72}a^{12}-\frac{45\!\cdots\!83}{24\!\cdots\!36}a^{11}+\frac{32\!\cdots\!79}{62\!\cdots\!09}a^{10}+\frac{53\!\cdots\!15}{24\!\cdots\!36}a^{9}-\frac{70\!\cdots\!73}{49\!\cdots\!72}a^{8}+\frac{78\!\cdots\!19}{49\!\cdots\!72}a^{7}+\frac{21\!\cdots\!63}{49\!\cdots\!72}a^{6}+\frac{11\!\cdots\!33}{24\!\cdots\!36}a^{5}-\frac{31\!\cdots\!11}{12\!\cdots\!18}a^{4}+\frac{15\!\cdots\!17}{49\!\cdots\!72}a^{3}-\frac{51\!\cdots\!28}{62\!\cdots\!09}a^{2}+\frac{94\!\cdots\!82}{62\!\cdots\!09}a+\frac{18\!\cdots\!36}{62\!\cdots\!09}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{4}$, which has order $4$ (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: | $\frac{15183643707763}{24\!\cdots\!48}a^{15}-\frac{6458624860247}{30\!\cdots\!31}a^{14}+\frac{29654455044491}{12\!\cdots\!24}a^{13}+\frac{448550656797}{30\!\cdots\!31}a^{12}-\frac{262634811485571}{24\!\cdots\!48}a^{11}-\frac{8336549560295}{60\!\cdots\!62}a^{10}+\frac{384620884517113}{30\!\cdots\!31}a^{9}-\frac{46\!\cdots\!67}{24\!\cdots\!48}a^{8}-\frac{144655654898357}{30\!\cdots\!31}a^{7}+\frac{12\!\cdots\!11}{60\!\cdots\!62}a^{6}-\frac{49\!\cdots\!29}{24\!\cdots\!48}a^{5}-\frac{476128750259418}{30\!\cdots\!31}a^{4}+\frac{10\!\cdots\!25}{24\!\cdots\!48}a^{3}+\frac{4515409868375}{24\!\cdots\!48}a^{2}+\frac{38956249626195}{12\!\cdots\!24}a+\frac{37\!\cdots\!69}{60\!\cdots\!62}$, $\frac{10\!\cdots\!73}{49\!\cdots\!72}a^{15}-\frac{23\!\cdots\!93}{24\!\cdots\!36}a^{14}+\frac{44\!\cdots\!61}{24\!\cdots\!36}a^{13}-\frac{10\!\cdots\!47}{62\!\cdots\!09}a^{12}-\frac{11\!\cdots\!75}{49\!\cdots\!72}a^{11}+\frac{85\!\cdots\!99}{24\!\cdots\!36}a^{10}+\frac{23\!\cdots\!54}{62\!\cdots\!09}a^{9}-\frac{53\!\cdots\!11}{49\!\cdots\!72}a^{8}+\frac{12\!\cdots\!55}{12\!\cdots\!18}a^{7}-\frac{22\!\cdots\!51}{12\!\cdots\!18}a^{6}-\frac{51\!\cdots\!49}{49\!\cdots\!72}a^{5}+\frac{32\!\cdots\!81}{24\!\cdots\!36}a^{4}+\frac{24\!\cdots\!71}{49\!\cdots\!72}a^{3}-\frac{66\!\cdots\!03}{49\!\cdots\!72}a^{2}+\frac{78\!\cdots\!25}{24\!\cdots\!36}a-\frac{62\!\cdots\!41}{12\!\cdots\!18}$, $\frac{75\!\cdots\!99}{49\!\cdots\!72}a^{15}-\frac{36\!\cdots\!76}{62\!\cdots\!09}a^{14}+\frac{64\!\cdots\!80}{62\!\cdots\!09}a^{13}-\frac{28\!\cdots\!47}{24\!\cdots\!36}a^{12}-\frac{75\!\cdots\!11}{49\!\cdots\!72}a^{11}+\frac{25\!\cdots\!79}{24\!\cdots\!36}a^{10}+\frac{33\!\cdots\!39}{12\!\cdots\!18}a^{9}-\frac{29\!\cdots\!81}{49\!\cdots\!72}a^{8}+\frac{13\!\cdots\!75}{24\!\cdots\!36}a^{7}-\frac{11\!\cdots\!93}{24\!\cdots\!36}a^{6}-\frac{39\!\cdots\!05}{49\!\cdots\!72}a^{5}+\frac{21\!\cdots\!39}{24\!\cdots\!36}a^{4}+\frac{37\!\cdots\!73}{49\!\cdots\!72}a^{3}+\frac{25\!\cdots\!81}{49\!\cdots\!72}a^{2}+\frac{75\!\cdots\!95}{24\!\cdots\!36}a+\frac{34\!\cdots\!49}{12\!\cdots\!18}$, $\frac{58\!\cdots\!49}{24\!\cdots\!36}a^{15}-\frac{23\!\cdots\!87}{24\!\cdots\!36}a^{14}+\frac{43\!\cdots\!39}{24\!\cdots\!36}a^{13}-\frac{41\!\cdots\!19}{24\!\cdots\!36}a^{12}-\frac{17\!\cdots\!83}{62\!\cdots\!09}a^{11}+\frac{36\!\cdots\!67}{12\!\cdots\!18}a^{10}+\frac{25\!\cdots\!73}{62\!\cdots\!09}a^{9}-\frac{25\!\cdots\!73}{24\!\cdots\!36}a^{8}+\frac{22\!\cdots\!59}{24\!\cdots\!36}a^{7}-\frac{85\!\cdots\!73}{24\!\cdots\!36}a^{6}-\frac{92\!\cdots\!91}{62\!\cdots\!09}a^{5}+\frac{11\!\cdots\!64}{62\!\cdots\!09}a^{4}+\frac{15\!\cdots\!23}{24\!\cdots\!36}a^{3}+\frac{63\!\cdots\!23}{62\!\cdots\!09}a^{2}+\frac{16\!\cdots\!86}{62\!\cdots\!09}a-\frac{13\!\cdots\!93}{62\!\cdots\!09}$, $\frac{86\!\cdots\!89}{12\!\cdots\!18}a^{15}-\frac{14\!\cdots\!13}{49\!\cdots\!72}a^{14}+\frac{29\!\cdots\!47}{49\!\cdots\!72}a^{13}-\frac{30\!\cdots\!55}{49\!\cdots\!72}a^{12}-\frac{17\!\cdots\!55}{49\!\cdots\!72}a^{11}+\frac{69\!\cdots\!39}{24\!\cdots\!36}a^{10}+\frac{32\!\cdots\!79}{24\!\cdots\!36}a^{9}-\frac{82\!\cdots\!97}{24\!\cdots\!36}a^{8}+\frac{14\!\cdots\!35}{49\!\cdots\!72}a^{7}-\frac{96\!\cdots\!37}{49\!\cdots\!72}a^{6}+\frac{71\!\cdots\!87}{49\!\cdots\!72}a^{5}+\frac{29\!\cdots\!59}{24\!\cdots\!36}a^{4}-\frac{49\!\cdots\!09}{24\!\cdots\!36}a^{3}+\frac{49\!\cdots\!23}{49\!\cdots\!72}a^{2}-\frac{31\!\cdots\!13}{24\!\cdots\!36}a-\frac{71\!\cdots\!69}{12\!\cdots\!18}$, $\frac{58\!\cdots\!97}{24\!\cdots\!36}a^{15}-\frac{49\!\cdots\!45}{49\!\cdots\!72}a^{14}+\frac{75\!\cdots\!33}{49\!\cdots\!72}a^{13}-\frac{41\!\cdots\!15}{49\!\cdots\!72}a^{12}-\frac{18\!\cdots\!49}{49\!\cdots\!72}a^{11}+\frac{46\!\cdots\!17}{12\!\cdots\!18}a^{10}+\frac{28\!\cdots\!54}{62\!\cdots\!09}a^{9}-\frac{64\!\cdots\!51}{62\!\cdots\!09}a^{8}+\frac{18\!\cdots\!49}{49\!\cdots\!72}a^{7}+\frac{14\!\cdots\!87}{49\!\cdots\!72}a^{6}-\frac{60\!\cdots\!03}{49\!\cdots\!72}a^{5}+\frac{16\!\cdots\!10}{62\!\cdots\!09}a^{4}+\frac{73\!\cdots\!37}{24\!\cdots\!36}a^{3}-\frac{56\!\cdots\!91}{49\!\cdots\!72}a^{2}+\frac{15\!\cdots\!85}{24\!\cdots\!36}a+\frac{18\!\cdots\!07}{12\!\cdots\!18}$, $\frac{29\!\cdots\!98}{62\!\cdots\!09}a^{15}-\frac{72\!\cdots\!26}{62\!\cdots\!09}a^{14}-\frac{47\!\cdots\!25}{24\!\cdots\!36}a^{13}+\frac{73\!\cdots\!47}{12\!\cdots\!18}a^{12}-\frac{13\!\cdots\!10}{62\!\cdots\!09}a^{11}+\frac{39\!\cdots\!55}{24\!\cdots\!36}a^{10}+\frac{19\!\cdots\!39}{24\!\cdots\!36}a^{9}-\frac{42\!\cdots\!85}{62\!\cdots\!09}a^{8}-\frac{60\!\cdots\!19}{24\!\cdots\!36}a^{7}+\frac{41\!\cdots\!77}{62\!\cdots\!09}a^{6}-\frac{83\!\cdots\!44}{62\!\cdots\!09}a^{5}+\frac{31\!\cdots\!61}{24\!\cdots\!36}a^{4}-\frac{92\!\cdots\!97}{24\!\cdots\!36}a^{3}-\frac{24\!\cdots\!12}{62\!\cdots\!09}a^{2}+\frac{60\!\cdots\!80}{62\!\cdots\!09}a-\frac{10\!\cdots\!31}{62\!\cdots\!09}$ (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: | \( 22132.6662656 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 22132.6662656 \cdot 4}{2\cdot\sqrt{19155446635260009765625}}\cr\approx \mathstrut & 0.776884642992 \end{aligned}\] (assuming GRH)
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
\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), 4.2.1375.1 x2, 4.0.15125.1 x2, \(\Q(\sqrt{5}, \sqrt{-11})\), 8.0.138403203125.2, 8.0.138403203125.1, 8.0.228765625.2 |
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: | 8.0.138403203125.1, 8.0.138403203125.2 |
Degree 16 sibling: | 16.4.19155446635260009765625.1 |
Minimal sibling: | 8.0.138403203125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
\(11\) | 11.8.6.1 | $x^{8} - 110 x^{4} - 16819$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
11.8.6.1 | $x^{8} - 110 x^{4} - 16819$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |