Number field 8.0.1327833.1

• Label: 8.0.1327833.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $1327833$
• Root discriminant: $5.83$
• Ramified primes: $3, 13, 97$
• Class number: $1$
• Class group: trivial
• Galois group: $C_2 \wr C_2\wr C_2$ (as 8T35)

Normalized defining polynomial

\( x^{8} - x^{7} + x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 1 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(1327833\)\(\medspace = 3^{4}\cdot 13^{2}\cdot 97\)
Root discriminant:		$5.83$
Ramified primes:		$3, 13, 97$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( 2 a^{7} - 2 a^{6} + a^{5} - 3 a^{4} + 6 a^{3} - 6 a^{2} + 5 a - 1 \) (order $6$)
Fundamental units:		\( a^{6} - a^{5} - 2 a^{3} + 3 a^{2} - 3 a + 2 \),  \( a^{7} - 2 a^{4} + a^{3} - a^{2} + a + 1 \),  \( a^{7} - a^{6} - 2 a^{4} + 3 a^{3} - 2 a^{2} + 2 a - 1 \)
Regulator:		\( 0.963083858701 \)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 0.963083858701 \cdot 1}{6\sqrt{1327833}}\approx 0.2171003873434$

Galois group

$D_4^2.C_2$ (as 8T35):

A solvable group of order 128

The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$

Character table for $C_2 \wr C_2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings:	data not computed
Degree 16 siblings:	data not computed
Degree 32 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	${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/5.8.0.1}{8} }$	${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.8.0.1}{8} }$	${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$3$	3.8.4.1	$x^{8} + 36 x^{4} - 27 x^{2} + 324$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$13$	13.4.0.1	$x^{4} + x^{2} - x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	13.4.2.1	$x^{4} + 39 x^{2} + 676$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
97	Data not computed

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
	1.291.2t1.a.a	$1$	$ 3 \cdot 97 $	\(\Q(\sqrt{-291}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.97.2t1.a.a	$1$	$ 97 $	\(\Q(\sqrt{97}) \)	$C_2$ (as 2T1)	$1$	$1$
*	1.3.2t1.a.a	$1$	$ 3 $	\(\Q(\sqrt{-3}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.3783.2t1.a.a	$1$	$ 3 \cdot 13 \cdot 97 $	\(\Q(\sqrt{-3783}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.13.2t1.a.a	$1$	$ 13 $	\(\Q(\sqrt{13}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.39.2t1.a.a	$1$	$ 3 \cdot 13 $	\(\Q(\sqrt{-39}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.1261.2t1.a.a	$1$	$ 13 \cdot 97 $	\(\Q(\sqrt{1261}) \)	$C_2$ (as 2T1)	$1$	$1$
	2.3783.4t3.b.a	$2$	$ 3 \cdot 13 \cdot 97 $	4.0.11349.1	$D_{4}$ (as 4T3)	$1$	$0$
	2.3783.4t3.a.a	$2$	$ 3 \cdot 13 \cdot 97 $	4.0.11349.2	$D_{4}$ (as 4T3)	$1$	$0$
*	2.39.4t3.a.a	$2$	$ 3 \cdot 13 $	4.0.117.1	$D_{4}$ (as 4T3)	$1$	$0$
	2.366951.4t3.a.a	$2$	$ 3 \cdot 13 \cdot 97^{2}$	4.0.1100853.1	$D_{4}$ (as 4T3)	$1$	$0$
	2.49179.4t3.b.a	$2$	$ 3 \cdot 13^{2} \cdot 97 $	4.0.147537.1	$D_{4}$ (as 4T3)	$1$	$0$
	2.291.4t3.c.a	$2$	$ 3 \cdot 97 $	4.0.873.1	$D_{4}$ (as 4T3)	$1$	$0$
	4.42933267.8t29.c.a	$4$	$ 3^{3} \cdot 13^{2} \cdot 97^{2}$	8.0.1211877327609.4	$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29)	$1$	$-2$
	4.1917981.8t35.b.a	$4$	$ 3^{2} \cdot 13^{3} \cdot 97 $	8.0.1327833.1	$C_2 \wr C_2\wr C_2$ (as 8T35)	$1$	$0$
	4.106782741.8t35.b.a	$4$	$ 3^{2} \cdot 13 \cdot 97^{3}$	8.0.1327833.1	$C_2 \wr C_2\wr C_2$ (as 8T35)	$1$	$0$
	4.4770363.8t29.c.a	$4$	$ 3 \cdot 13^{2} \cdot 97^{2}$	8.0.1211877327609.4	$(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29)	$1$	$2$
	4.18046283229.8t35.b.a	$4$	$ 3^{2} \cdot 13^{3} \cdot 97^{3}$	8.0.1327833.1	$C_2 \wr C_2\wr C_2$ (as 8T35)	$1$	$0$
*	4.11349.8t35.b.a	$4$	$ 3^{2} \cdot 13 \cdot 97 $	8.0.1327833.1	$C_2 \wr C_2\wr C_2$ (as 8T35)	$1$	$0$

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 *.