Global number field 8.0.2420000000.4

• Label: 8.0.2420000000.4
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{8}\cdot 5^{7}\cdot 11^{2}$
• Root discriminant: $14.89$
• Ramified primes: $2, 5, 11$
• Class number: $2$
• Class group: $[2]$
• Galois group: $(C_8:C_2):C_2$ (as 8T16)

Normalized defining polynomial

\( x^{8} + 50 x^{4} + 605 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(2420000000=2^{8}\cdot 5^{7}\cdot 11^{2}\)
Root discriminant:		$14.89$
Ramified primes:		$2, 5, 11$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{88} a^{6} - \frac{1}{8} a^{4} - \frac{5}{88} a^{2} - \frac{3}{8}$, $\frac{1}{88} a^{7} - \frac{1}{8} a^{5} - \frac{5}{88} a^{3} - \frac{3}{8} a$

Class group and class number

$C_{2}$, which has order $2$

Unit group

Rank:		$3$
Torsion generator:		\( \frac{3}{88} a^{6} + \frac{1}{8} a^{4} + \frac{73}{88} a^{2} + \frac{27}{8} \) (order $10$)
Fundamental units:		\( \frac{1}{88} a^{6} + \frac{1}{8} a^{4} + \frac{39}{88} a^{2} + \frac{31}{8} \),  \( \frac{5}{88} a^{7} + \frac{5}{44} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{151}{88} a^{3} + \frac{151}{44} a^{2} + \frac{27}{8} a - \frac{31}{4} \),  \( \frac{7}{88} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{7}{8} a^{4} + \frac{141}{88} a^{3} - \frac{19}{8} a^{2} + \frac{27}{8} a + \frac{157}{8} \)
Regulator:		\( 57.1742004281 \)

Galois group

$C_2^3.C_4$ (as 8T16):

A solvable group of order 32

The 11 conjugacy class representatives for $(C_8:C_2):C_2$

Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\)

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 sibling:	data not computed
Degree 16 siblings:	data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.8.0.1}{8} }$	R	${\href{/LocalNumberField/7.8.0.1}{8} }$	R	${\href{/LocalNumberField/13.8.0.1}{8} }$	${\href{/LocalNumberField/17.8.0.1}{8} }$	${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.8.0.1}{8} }$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$	2.8.8.6	$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$	$2$	$4$	$8$	$(C_8:C_2):C_2$	$[2, 2, 2]^{4}$
$5$	5.8.7.2	$x^{8} - 20$	$8$	$1$	$7$	$C_8:C_2$	$[\ ]_{8}^{2}$
$11$	$\Q_{11}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{11}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.1	$x^{2} - 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.0.1	$x^{2} - x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e2_5.2t1.1c1	$1$	$ 2^{2} \cdot 5 $	$x^{2} + 5$	$C_2$ (as 2T1)	$1$	$-1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_5.4t1.1c1	$1$	$ 2^{2} \cdot 5 $	$x^{4} - 5 x^{2} + 5$	$C_4$ (as 4T1)	$0$	$1$
*	1.5.4t1.1c1	$1$	$ 5 $	$x^{4} - x^{3} + x^{2} - x + 1$	$C_4$ (as 4T1)	$0$	$-1$
	1.2e2_5.4t1.1c2	$1$	$ 2^{2} \cdot 5 $	$x^{4} - 5 x^{2} + 5$	$C_4$ (as 4T1)	$0$	$1$
*	1.5.4t1.1c2	$1$	$ 5 $	$x^{4} - x^{3} + x^{2} - x + 1$	$C_4$ (as 4T1)	$0$	$-1$
	2.2e4_5_11e2.4t3.6c1	$2$	$ 2^{4} \cdot 5 \cdot 11^{2}$	$x^{4} - 2 x^{3} - 20 x^{2} + 10 x + 157$	$D_{4}$ (as 4T3)	$1$	$0$
	2.2e4_5e2_11e2.4t3.3c1	$2$	$ 2^{4} \cdot 5^{2} \cdot 11^{2}$	$x^{4} - 2 x^{3} + 4 x^{2} + 52 x + 126$	$D_{4}$ (as 4T3)	$1$	$0$
*	4.2e8_5e4_11e2.8t16.9c1	$4$	$ 2^{8} \cdot 5^{4} \cdot 11^{2}$	$x^{8} + 50 x^{4} + 605$	$(C_8:C_2):C_2$ (as 8T16)	$1$	$0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.