Normalized defining polynomial
\( x^{8} - 2x^{7} - 19x^{6} + 26x^{5} + 91x^{4} - 108x^{3} - 114x^{2} + 144x - 36 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(27710263296\) \(\medspace = 2^{12}\cdot 3^{4}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.20\) | 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: | $2^{3/2}3^{1/2}17^{1/2}\approx 20.199009876724155$ | ||
Ramified primes: | \(2\), \(3\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{546}a^{7}-\frac{2}{91}a^{6}+\frac{5}{273}a^{5}+\frac{18}{91}a^{4}-\frac{79}{546}a^{3}+\frac{227}{546}a^{2}-\frac{3}{91}a-\frac{37}{91}$
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: | $\frac{11}{39}a^{7}-\frac{5}{13}a^{6}-\frac{443}{78}a^{5}+\frac{103}{26}a^{4}+\frac{2279}{78}a^{3}-\frac{584}{39}a^{2}-\frac{560}{13}a+\frac{265}{13}$, $\frac{5}{14}a^{7}-\frac{19}{42}a^{6}-\frac{149}{21}a^{5}+\frac{89}{21}a^{4}+\frac{734}{21}a^{3}-\frac{655}{42}a^{2}-\frac{346}{7}a+\frac{152}{7}$, $\frac{93}{182}a^{7}-\frac{218}{273}a^{6}-\frac{5491}{546}a^{5}+\frac{2417}{273}a^{4}+\frac{27463}{546}a^{3}-\frac{8783}{273}a^{2}-\frac{6570}{91}a+\frac{3509}{91}$, $\frac{11}{182}a^{7}-\frac{16}{273}a^{6}-\frac{671}{546}a^{5}+\frac{197}{546}a^{4}+\frac{1654}{273}a^{3}-\frac{335}{546}a^{2}-\frac{645}{91}a+\frac{235}{91}$, $\frac{20}{39}a^{7}-\frac{32}{39}a^{6}-\frac{261}{26}a^{5}+\frac{353}{39}a^{4}+\frac{652}{13}a^{3}-\frac{2477}{78}a^{2}-\frac{939}{13}a+\frac{457}{13}$, $\frac{51}{182}a^{7}-\frac{33}{91}a^{6}-\frac{1037}{182}a^{5}+\frac{685}{182}a^{4}+\frac{2672}{91}a^{3}-\frac{2801}{182}a^{2}-\frac{3917}{91}a+\frac{2074}{91}$, $\frac{47}{273}a^{7}-\frac{127}{546}a^{6}-\frac{627}{182}a^{5}+\frac{617}{273}a^{4}+\frac{3197}{182}a^{3}-\frac{3869}{546}a^{2}-\frac{2375}{91}a+\frac{617}{91}$ | 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: | \( 494.438755813 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 494.438755813 \cdot 1}{2\cdot\sqrt{27710263296}}\cr\approx \mathstrut & 0.380191277057 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 4.4.20808.1 x2, 4.4.9792.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.4.20808.1, 4.4.9792.1 |
Minimal sibling: | 4.4.9792.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(17\) | 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.136.2t1.a.a | $1$ | $ 2^{3} \cdot 17 $ | \(\Q(\sqrt{34}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.1224.4t3.a.a | $2$ | $ 2^{3} \cdot 3^{2} \cdot 17 $ | 8.8.27710263296.1 | $D_4$ (as 8T4) | $1$ | $2$ |