Normalized defining polynomial
\( x^{8} - 12x^{6} + 28x^{4} + 104x^{2} + 169 \)
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)
| |
Discriminant: | \(11341398016\) \(\medspace = 2^{26}\cdot 13^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.06\) | 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: | $2^{61/16}13^{1/2}\approx 50.658057734962696$ | ||
Ramified primes: | \(2\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3731}a^{6}+\frac{1496}{3731}a^{4}-\frac{1259}{3731}a^{2}+\frac{47}{287}$, $\frac{1}{3731}a^{7}+\frac{1496}{3731}a^{5}-\frac{1259}{3731}a^{3}+\frac{47}{287}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{4}{287} a^{6} - \frac{43}{287} a^{4} + \frac{130}{287} a^{2} + \frac{148}{287} \) (order $4$) | 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: | $\frac{89}{3731}a^{6}-\frac{1172}{3731}a^{4}+\frac{3610}{3731}a^{2}+\frac{452}{287}$, $\frac{57}{3731}a^{7}-\frac{1}{287}a^{6}-\frac{541}{3731}a^{5}-\frac{61}{287}a^{4}+\frac{2857}{3731}a^{3}+\frac{111}{287}a^{2}+\frac{383}{287}a-\frac{324}{287}$, $\frac{548}{3731}a^{7}-\frac{54}{533}a^{6}-\frac{8474}{3731}a^{5}+\frac{232}{533}a^{4}+\frac{33882}{3731}a^{3}+\frac{1894}{533}a^{2}+\frac{1648}{287}a-\frac{775}{41}$ (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: | \( 206.862391313 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 206.862391313 \cdot 1}{4\cdot\sqrt{11341398016}}\cr\approx \mathstrut & 0.756846360687 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3:C_4$ (as 8T19):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3 : C_4 $ |
Character table for $C_2^3 : C_4 $ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 4.0.512.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.26.71 | $x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 24 x^{2} + 2$ | $8$ | $1$ | $26$ | $C_2^3 : C_4 $ | $[2, 3, 7/2, 4, 17/4]$ |
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.208.4t1.f.a | $1$ | $ 2^{4} \cdot 13 $ | 4.0.346112.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.208.4t1.e.a | $1$ | $ 2^{4} \cdot 13 $ | 4.4.346112.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.208.4t1.e.b | $1$ | $ 2^{4} \cdot 13 $ | 4.4.346112.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.208.4t1.f.b | $1$ | $ 2^{4} \cdot 13 $ | 4.0.346112.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 2.128.4t3.a.a | $2$ | $ 2^{7}$ | 4.0.512.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.43264.4t3.d.a | $2$ | $ 2^{8} \cdot 13^{2}$ | 4.0.346112.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.22151168.8t19.a.a | $4$ | $ 2^{17} \cdot 13^{2}$ | 8.0.11341398016.3 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |