Normalized defining polynomial
\( x^{8} - 2x^{7} + 3x^{6} - 8x^{5} + 28x^{4} - 42x^{3} + 11x^{2} + 20x + 5 \)
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: | \(4569760000\) \(\medspace = 2^{8}\cdot 5^{4}\cdot 13^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.12\) | 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: | $2\cdot 5^{3/4}13^{3/4}\approx 45.78413496570207$ | ||
Ramified primes: | \(2\), \(5\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{1688}a^{7}-\frac{171}{1688}a^{6}+\frac{103}{844}a^{5}+\frac{313}{844}a^{4}+\frac{289}{844}a^{3}+\frac{45}{422}a^{2}-\frac{25}{1688}a-\frac{819}{1688}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{69}{844} a^{7} - \frac{97}{422} a^{6} + \frac{72}{211} a^{5} - \frac{347}{422} a^{4} + \frac{581}{211} a^{3} - \frac{1115}{211} a^{2} + \frac{2495}{844} a + \frac{273}{211} \) (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: | $\frac{99}{1688}a^{7}-\frac{49}{1688}a^{6}+\frac{69}{844}a^{5}-\frac{241}{844}a^{4}+\frac{759}{844}a^{3}-\frac{187}{422}a^{2}-\frac{2475}{1688}a-\frac{57}{1688}$, $\frac{5}{422}a^{7}-\frac{11}{422}a^{6}-\frac{25}{422}a^{5}-\frac{35}{422}a^{4}+\frac{147}{422}a^{3}+\frac{267}{422}a^{2}-\frac{168}{211}a-\frac{43}{211}$, $\frac{145}{1688}a^{7}+\frac{103}{1688}a^{6}+\frac{165}{844}a^{5}-\frac{191}{844}a^{4}+\frac{971}{844}a^{3}+\frac{195}{422}a^{2}-\frac{1937}{1688}a-\frac{173}{1688}$ | 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: | \( 52.0946915903 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 52.0946915903 \cdot 2}{4\cdot\sqrt{4569760000}}\cr\approx \mathstrut & 0.600532136746 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), 4.0.1040.1 |
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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.3.4 | $x^{4} + 91$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
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) | $1$ | $-1$ |
1.260.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 13 $ | \(\Q(\sqrt{-65}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.65.2t1.a.a | $1$ | $ 5 \cdot 13 $ | \(\Q(\sqrt{65}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.260.4t1.c.a | $1$ | $ 2^{2} \cdot 5 \cdot 13 $ | 4.0.4394000.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.65.4t1.d.a | $1$ | $ 5 \cdot 13 $ | 4.4.274625.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.65.4t1.d.b | $1$ | $ 5 \cdot 13 $ | 4.4.274625.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.260.4t1.c.b | $1$ | $ 2^{2} \cdot 5 \cdot 13 $ | 4.0.4394000.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 2.260.4t3.b.a | $2$ | $ 2^{2} \cdot 5 \cdot 13 $ | 4.0.1040.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.16900.4t3.b.a | $2$ | $ 2^{2} \cdot 5^{2} \cdot 13^{2}$ | 4.0.4394000.3 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.4394000.8t19.a.a | $4$ | $ 2^{4} \cdot 5^{3} \cdot 13^{3}$ | 8.0.4569760000.7 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |