Normalized defining polynomial
\( x^{16} + 2x^{14} - 14x^{10} + 49x^{4} - 26x^{2} + 4 \)
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: | \(11378694989371408384\) \(\medspace = 2^{24}\cdot 7^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.52\) | 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: | $2^{3/2}7^{7/8}\approx 15.524076653923595$ | ||
Ramified primes: | \(2\), \(7\) | 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)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{20}a^{10}-\frac{1}{10}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{9}{20}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{2}{5}$, $\frac{1}{20}a^{11}-\frac{1}{10}a^{9}-\frac{1}{20}a^{5}-\frac{1}{2}a^{2}+\frac{2}{5}a$, $\frac{1}{20}a^{12}-\frac{1}{5}a^{8}-\frac{1}{20}a^{6}-\frac{1}{10}a^{4}-\frac{1}{2}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{20}a^{13}-\frac{1}{5}a^{9}-\frac{1}{20}a^{7}-\frac{1}{10}a^{5}-\frac{1}{2}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{260}a^{14}-\frac{1}{130}a^{12}-\frac{1}{52}a^{10}+\frac{19}{260}a^{8}-\frac{5}{26}a^{6}+\frac{83}{260}a^{4}-\frac{1}{2}a^{3}+\frac{6}{13}a^{2}+\frac{2}{13}$, $\frac{1}{520}a^{15}-\frac{1}{520}a^{14}+\frac{11}{520}a^{13}-\frac{11}{520}a^{12}-\frac{1}{104}a^{11}+\frac{1}{104}a^{10}+\frac{97}{520}a^{9}-\frac{97}{520}a^{8}-\frac{63}{520}a^{7}+\frac{63}{520}a^{6}+\frac{57}{520}a^{5}-\frac{57}{520}a^{4}+\frac{47}{260}a^{3}-\frac{47}{260}a^{2}-\frac{3}{130}a+\frac{3}{130}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{11}{104}a^{15}-\frac{1}{8}a^{14}+\frac{137}{520}a^{13}-\frac{11}{40}a^{12}+\frac{63}{520}a^{11}-\frac{1}{40}a^{10}-\frac{749}{520}a^{9}+\frac{71}{40}a^{8}-\frac{397}{520}a^{7}+\frac{11}{40}a^{6}-\frac{167}{520}a^{5}-\frac{7}{40}a^{4}+\frac{1363}{260}a^{3}-\frac{119}{20}a^{2}-\frac{61}{130}a+\frac{19}{10}$, $\frac{11}{104}a^{15}+\frac{1}{8}a^{14}+\frac{137}{520}a^{13}+\frac{11}{40}a^{12}+\frac{63}{520}a^{11}+\frac{1}{40}a^{10}-\frac{749}{520}a^{9}-\frac{71}{40}a^{8}-\frac{397}{520}a^{7}-\frac{11}{40}a^{6}-\frac{167}{520}a^{5}+\frac{7}{40}a^{4}+\frac{1363}{260}a^{3}+\frac{119}{20}a^{2}-\frac{61}{130}a-\frac{19}{10}$, $\frac{15}{104}a^{15}+\frac{83}{520}a^{14}+\frac{35}{104}a^{13}+\frac{37}{104}a^{12}+\frac{41}{520}a^{11}+\frac{53}{520}a^{10}-\frac{1097}{520}a^{9}-\frac{1153}{520}a^{8}-\frac{87}{104}a^{7}-\frac{341}{520}a^{6}-\frac{41}{520}a^{5}-\frac{261}{520}a^{4}+\frac{367}{52}a^{3}+\frac{2159}{260}a^{2}-\frac{173}{130}a-\frac{55}{26}$, $\frac{23}{130}a^{15}-\frac{1}{65}a^{14}+\frac{103}{260}a^{13}-\frac{1}{52}a^{12}+\frac{17}{260}a^{11}-\frac{3}{130}a^{10}-\frac{33}{13}a^{9}+\frac{7}{65}a^{8}-\frac{31}{52}a^{7}-\frac{47}{260}a^{6}+\frac{61}{260}a^{5}+\frac{11}{26}a^{4}+\frac{227}{26}a^{3}-\frac{97}{130}a^{2}-\frac{216}{65}a+\frac{51}{65}$, $\frac{7}{260}a^{15}-\frac{1}{130}a^{14}-\frac{1}{260}a^{13}+\frac{1}{65}a^{12}-\frac{12}{65}a^{11}+\frac{23}{260}a^{10}-\frac{153}{260}a^{9}+\frac{33}{130}a^{8}+\frac{157}{260}a^{7}-\frac{3}{26}a^{6}+\frac{89}{130}a^{5}-\frac{49}{260}a^{4}+\frac{277}{130}a^{3}-\frac{37}{26}a^{2}-\frac{164}{65}a+\frac{6}{65}$, $\frac{1}{26}a^{15}+\frac{7}{130}a^{14}+\frac{19}{260}a^{13}+\frac{37}{260}a^{12}-\frac{11}{260}a^{11}+\frac{21}{260}a^{10}-\frac{87}{130}a^{9}-\frac{44}{65}a^{8}-\frac{19}{260}a^{7}-\frac{23}{52}a^{6}+\frac{63}{260}a^{5}-\frac{99}{260}a^{4}+\frac{301}{130}a^{3}+\frac{32}{13}a^{2}-\frac{56}{65}a-\frac{3}{65}$, $\frac{1}{26}a^{15}-\frac{7}{130}a^{14}+\frac{19}{260}a^{13}-\frac{37}{260}a^{12}-\frac{11}{260}a^{11}-\frac{21}{260}a^{10}-\frac{87}{130}a^{9}+\frac{44}{65}a^{8}-\frac{19}{260}a^{7}+\frac{23}{52}a^{6}+\frac{63}{260}a^{5}+\frac{99}{260}a^{4}+\frac{301}{130}a^{3}-\frac{32}{13}a^{2}-\frac{56}{65}a+\frac{3}{65}$ | 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: | \( 1569.86019374 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1569.86019374 \cdot 1}{2\cdot\sqrt{11378694989371408384}}\cr\approx \mathstrut & 0.565228042859 \end{aligned}\]
Galois group
A solvable group of order 16 |
The 7 conjugacy class representatives for $D_{8}$ |
Character table for $D_{8}$ |
Intermediate fields
\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-2}, \sqrt{-7})\), 4.0.2744.1 x2, 4.2.21952.1 x2, 8.0.481890304.3, 8.0.421654016.1 x4, 8.2.3373232128.2 x4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(7\) | 7.16.14.1 | $x^{16} + 28 x^{9} + 14 x^{8} - 98 x^{2} + 196 x + 49$ | $8$ | $2$ | $14$ | $D_{8}$ | $[\ ]_{8}^{2}$ |