Normalized defining polynomial
\( x^{8} - 52x^{6} - 490x^{5} - 242x^{4} + 12740x^{3} + 83893x^{2} + 224910x + 297036 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(8475784699200625\) \(\medspace = 5^{4}\cdot 19^{4}\cdot 101^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(97.95\) | 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: | $5^{1/2}19^{1/2}101^{1/2}\approx 97.95407086997457$ | ||
Ramified primes: | \(5\), \(19\), \(101\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6}a^{4}-\frac{1}{3}a^{2}+\frac{1}{6}a$, $\frac{1}{6}a^{5}-\frac{1}{3}a^{3}+\frac{1}{6}a^{2}$, $\frac{1}{42}a^{6}+\frac{1}{14}a^{4}+\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a+\frac{2}{7}$, $\frac{1}{1489102482}a^{7}-\frac{3678989}{1489102482}a^{6}-\frac{47564393}{744551241}a^{5}-\frac{21328834}{744551241}a^{4}+\frac{37380863}{212728926}a^{3}-\frac{4770657}{35454821}a^{2}+\frac{269069393}{1489102482}a-\frac{62265964}{248183747}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{12}\times C_{96}$, which has order $1152$
Unit group
Rank: | $3$ | 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: | $\frac{23876}{248183747}a^{7}-\frac{444673}{496367494}a^{6}+\frac{90277}{744551241}a^{5}-\frac{3131423}{496367494}a^{4}+\frac{43110551}{212728926}a^{3}-\frac{29783609}{106364463}a^{2}-\frac{1111846389}{496367494}a-\frac{2943638000}{248183747}$, $\frac{7048}{248183747}a^{7}-\frac{666142}{744551241}a^{6}+\frac{1417585}{496367494}a^{5}+\frac{5638016}{248183747}a^{4}+\frac{22097540}{106364463}a^{3}-\frac{198944461}{212728926}a^{2}-\frac{3883726841}{744551241}a-\frac{4163563303}{248183747}$, $\frac{994}{106364463}a^{7}-\frac{24700}{744551241}a^{6}-\frac{5677}{106364463}a^{5}-\frac{4039885}{1489102482}a^{4}-\frac{174034}{35454821}a^{3}-\frac{15658879}{106364463}a^{2}-\frac{138750427}{212728926}a-\frac{438242457}{248183747}$ | 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: | \( 114.601534853 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 114.601534853 \cdot 1152}{2\cdot\sqrt{8475784699200625}}\cr\approx \mathstrut & 1.11748687981 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{-9595}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-95}, \sqrt{101})\), 4.2.969095.1 x2, 4.0.911525.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.2.969095.1, 4.0.911525.1 |
Minimal sibling: | 4.0.911525.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.1.0.1}{1} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(19\) | 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(101\) | 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.95.2t1.a.a | $1$ | $ 5 \cdot 19 $ | \(\Q(\sqrt{-95}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.101.2t1.a.a | $1$ | $ 101 $ | \(\Q(\sqrt{101}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.9595.2t1.a.a | $1$ | $ 5 \cdot 19 \cdot 101 $ | \(\Q(\sqrt{-9595}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
*2 | 2.9595.4t3.r.a | $2$ | $ 5 \cdot 19 \cdot 101 $ | 8.0.8475784699200625.6 | $D_4$ (as 8T4) | $1$ | $0$ |