Number field 18.4.14616700658977641856215848806693687375.1

• Label: 18.4.146...375.1
• Degree: $18$
• Signature: $[4, 7]$
• Discriminant: $-1.462\times 10^{37}$
• Root discriminant: $116.07$
• Ramified primes: $5,7,41,419,449$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_3^6.C_2\wr C_6$ (as 18T858)

Normalized defining polynomial

x^18 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125

Invariants

Degree:		$18$
Signature:		$[4, 7]$
Discriminant:		$-14616700658977641856215848806693687375$ $\medspace = -\,5^{3}\cdot 7^{12}\cdot 41^{4}\cdot 419^{3}\cdot 449^{4}$
Root discriminant:		$116.07$
Galois root discriminant:		$5^{1/2}7^{2/3}41^{2/3}419^{1/2}449^{2/3}\approx 116773.46634277423$
Ramified primes:		$5$, $7$, $41$, $419$, $449$
Discriminant root field:		$\Q(\sqrt{-2095})$
$\Aut(K/\Q)$:		$C_1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{105}a^{16}-\frac{4}{105}a^{15}+\frac{22}{105}a^{14}+\frac{3}{7}a^{13}-\frac{1}{3}a^{12}-\frac{26}{105}a^{11}+\frac{4}{105}a^{10}+\frac{1}{5}a^{9}+\frac{44}{105}a^{8}-\frac{19}{105}a^{7}+\frac{1}{21}a^{6}+\frac{4}{15}a^{5}-\frac{19}{105}a^{4}+\frac{52}{105}a^{3}-\frac{8}{21}a^{2}-\frac{11}{35}a-\frac{5}{21}$, $\frac{1}{12\cdots 75}a^{17}+\frac{52\cdots 26}{12\cdots 75}a^{16}-\frac{47\cdots 84}{18\cdots 25}a^{15}-\frac{93\cdots 46}{25\cdots 95}a^{14}+\frac{57\cdots 28}{25\cdots 95}a^{13}-\frac{10\cdots 04}{14\cdots 75}a^{12}-\frac{19\cdots 77}{42\cdots 25}a^{11}-\frac{53\cdots 54}{12\cdots 75}a^{10}-\frac{26\cdots 71}{12\cdots 75}a^{9}+\frac{35\cdots 22}{42\cdots 25}a^{8}+\frac{13\cdots 72}{28\cdots 55}a^{7}-\frac{54\cdots 82}{12\cdots 75}a^{6}+\frac{35\cdots 76}{12\cdots 75}a^{5}-\frac{57\cdots 34}{18\cdots 25}a^{4}+\frac{13\cdots 07}{25\cdots 95}a^{3}-\frac{49\cdots 68}{12\cdots 75}a^{2}-\frac{89\cdots 58}{36\cdots 85}a-\frac{14\cdots 03}{50\cdots 19}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)
Narrow class group:		Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$10$
Torsion generator:		$-1$  (order $2$)
Fundamental units:		$\frac{59\cdots 24}{76\cdots 25}a^{17}-\frac{65\cdots 76}{76\cdots 25}a^{16}-\frac{32\cdots 12}{76\cdots 25}a^{15}+\frac{41\cdots 51}{15\cdots 65}a^{14}-\frac{13\cdots 23}{15\cdots 65}a^{13}-\frac{11\cdots 88}{25\cdots 75}a^{12}-\frac{12\cdots 48}{25\cdots 75}a^{11}+\frac{31\cdots 04}{76\cdots 25}a^{10}+\frac{27\cdots 21}{76\cdots 25}a^{9}-\frac{17\cdots 74}{85\cdots 25}a^{8}-\frac{27\cdots 63}{73\cdots 65}a^{7}+\frac{23\cdots 32}{76\cdots 25}a^{6}+\frac{10\cdots 74}{76\cdots 25}a^{5}+\frac{67\cdots 38}{76\cdots 25}a^{4}-\frac{11\cdots 67}{15\cdots 65}a^{3}-\frac{10\cdots 32}{76\cdots 25}a^{2}-\frac{11\cdots 69}{15\cdots 65}a-\frac{56\cdots 70}{30\cdots 53}$, $\frac{59\cdots 24}{76\cdots 25}a^{17}-\frac{65\cdots 76}{76\cdots 25}a^{16}-\frac{32\cdots 12}{76\cdots 25}a^{15}+\frac{41\cdots 51}{15\cdots 65}a^{14}-\frac{13\cdots 23}{15\cdots 65}a^{13}-\frac{11\cdots 88}{25\cdots 75}a^{12}-\frac{12\cdots 48}{25\cdots 75}a^{11}+\frac{31\cdots 04}{76\cdots 25}a^{10}+\frac{27\cdots 21}{76\cdots 25}a^{9}-\frac{17\cdots 74}{85\cdots 25}a^{8}-\frac{27\cdots 63}{73\cdots 65}a^{7}+\frac{23\cdots 32}{76\cdots 25}a^{6}+\frac{10\cdots 74}{76\cdots 25}a^{5}+\frac{67\cdots 38}{76\cdots 25}a^{4}-\frac{11\cdots 67}{15\cdots 65}a^{3}-\frac{10\cdots 32}{76\cdots 25}a^{2}-\frac{11\cdots 69}{15\cdots 65}a-\frac{25\cdots 17}{30\cdots 53}$, $\frac{39\cdots 24}{12\cdots 75}a^{17}-\frac{47\cdots 76}{12\cdots 75}a^{16}+\frac{36\cdots 88}{12\cdots 75}a^{15}+\frac{23\cdots 41}{25\cdots 95}a^{14}-\frac{29\cdots 68}{25\cdots 95}a^{13}-\frac{55\cdots 38}{42\cdots 25}a^{12}+\frac{88\cdots 84}{14\cdots 75}a^{11}+\frac{14\cdots 79}{12\cdots 75}a^{10}+\frac{85\cdots 21}{12\cdots 75}a^{9}-\frac{20\cdots 47}{42\cdots 25}a^{8}-\frac{10\cdots 48}{12\cdots 95}a^{7}+\frac{39\cdots 57}{12\cdots 75}a^{6}+\frac{31\cdots 74}{12\cdots 75}a^{5}+\frac{38\cdots 13}{12\cdots 75}a^{4}+\frac{41\cdots 13}{25\cdots 95}a^{3}+\frac{24\cdots 43}{12\cdots 75}a^{2}-\frac{27\cdots 34}{25\cdots 95}a+\frac{27\cdots 50}{50\cdots 19}$, $\frac{19\cdots 67}{14\cdots 75}a^{17}-\frac{22\cdots 28}{14\cdots 75}a^{16}+\frac{20\cdots 09}{14\cdots 75}a^{15}+\frac{21\cdots 33}{56\cdots 91}a^{14}-\frac{12\cdots 14}{28\cdots 55}a^{13}-\frac{70\cdots 87}{14\cdots 75}a^{12}+\frac{17\cdots 68}{14\cdots 75}a^{11}+\frac{57\cdots 02}{14\cdots 75}a^{10}+\frac{60\cdots 48}{14\cdots 75}a^{9}-\frac{19\cdots 08}{14\cdots 75}a^{8}-\frac{16\cdots 44}{40\cdots 65}a^{7}-\frac{28\cdots 19}{14\cdots 75}a^{6}+\frac{11\cdots 32}{14\cdots 75}a^{5}+\frac{25\cdots 09}{14\cdots 75}a^{4}+\frac{53\cdots 91}{28\cdots 55}a^{3}+\frac{16\cdots 69}{14\cdots 75}a^{2}+\frac{25\cdots 69}{56\cdots 91}a+\frac{48\cdots 37}{56\cdots 91}$, $\frac{26\cdots 99}{25\cdots 95}a^{17}-\frac{52\cdots 94}{50\cdots 19}a^{16}-\frac{42\cdots 06}{25\cdots 95}a^{15}+\frac{86\cdots 27}{25\cdots 95}a^{14}+\frac{13\cdots 40}{50\cdots 19}a^{13}-\frac{45\cdots 73}{84\cdots 65}a^{12}-\frac{40\cdots 30}{56\cdots 91}a^{11}+\frac{11\cdots 93}{25\cdots 95}a^{10}+\frac{26\cdots 57}{25\cdots 95}a^{9}-\frac{16\cdots 12}{12\cdots 95}a^{8}-\frac{55\cdots 58}{84\cdots 65}a^{7}-\frac{10\cdots 33}{25\cdots 95}a^{6}+\frac{30\cdots 07}{25\cdots 95}a^{5}+\frac{65\cdots 09}{25\cdots 95}a^{4}+\frac{57\cdots 92}{25\cdots 95}a^{3}+\frac{41\cdots 59}{36\cdots 85}a^{2}+\frac{92\cdots 57}{25\cdots 95}a+\frac{16\cdots 92}{50\cdots 19}$, $\frac{15\cdots 46}{12\cdots 75}a^{17}-\frac{15\cdots 39}{12\cdots 75}a^{16}-\frac{21\cdots 83}{12\cdots 75}a^{15}+\frac{19\cdots 41}{50\cdots 19}a^{14}+\frac{90\cdots 59}{36\cdots 85}a^{13}-\frac{85\cdots 09}{14\cdots 75}a^{12}-\frac{30\cdots 47}{42\cdots 25}a^{11}+\frac{92\cdots 93}{18\cdots 25}a^{10}+\frac{14\cdots 49}{12\cdots 75}a^{9}-\frac{70\cdots 68}{42\cdots 25}a^{8}-\frac{21\cdots 21}{28\cdots 55}a^{7}-\frac{77\cdots 21}{18\cdots 25}a^{6}+\frac{19\cdots 16}{12\cdots 75}a^{5}+\frac{40\cdots 17}{12\cdots 75}a^{4}+\frac{55\cdots 28}{25\cdots 95}a^{3}-\frac{20\cdots 28}{12\cdots 75}a^{2}-\frac{55\cdots 52}{50\cdots 19}a-\frac{37\cdots 32}{72\cdots 17}$, $\frac{53\cdots 62}{60\cdots 75}a^{17}-\frac{44\cdots 91}{42\cdots 25}a^{16}+\frac{39\cdots 83}{42\cdots 25}a^{15}+\frac{70\cdots 97}{28\cdots 55}a^{14}-\frac{24\cdots 48}{84\cdots 65}a^{13}-\frac{20\cdots 82}{60\cdots 75}a^{12}+\frac{41\cdots 71}{42\cdots 25}a^{11}+\frac{40\cdots 63}{14\cdots 75}a^{10}+\frac{15\cdots 48}{60\cdots 75}a^{9}-\frac{47\cdots 31}{42\cdots 25}a^{8}-\frac{22\cdots 63}{84\cdots 65}a^{7}-\frac{18\cdots 38}{42\cdots 25}a^{6}+\frac{34\cdots 12}{60\cdots 75}a^{5}+\frac{44\cdots 33}{42\cdots 25}a^{4}+\frac{84\cdots 28}{84\cdots 65}a^{3}+\frac{80\cdots 71}{14\cdots 75}a^{2}+\frac{17\cdots 16}{84\cdots 65}a+\frac{23\cdots 26}{56\cdots 91}$, $\frac{27\cdots 29}{12\cdots 75}a^{17}-\frac{34\cdots 46}{12\cdots 75}a^{16}+\frac{38\cdots 48}{12\cdots 75}a^{15}+\frac{16\cdots 71}{25\cdots 95}a^{14}-\frac{31\cdots 18}{25\cdots 95}a^{13}-\frac{37\cdots 73}{42\cdots 25}a^{12}+\frac{48\cdots 42}{42\cdots 25}a^{11}+\frac{95\cdots 09}{12\cdots 75}a^{10}-\frac{27\cdots 34}{12\cdots 75}a^{9}-\frac{51\cdots 79}{14\cdots 75}a^{8}-\frac{32\cdots 73}{12\cdots 95}a^{7}+\frac{84\cdots 72}{12\cdots 75}a^{6}+\frac{16\cdots 54}{12\cdots 75}a^{5}+\frac{88\cdots 23}{12\cdots 75}a^{4}-\frac{72\cdots 57}{25\cdots 95}a^{3}-\frac{80\cdots 22}{12\cdots 75}a^{2}-\frac{87\cdots 49}{25\cdots 95}a-\frac{67\cdots 97}{50\cdots 19}$, $\frac{19\cdots 57}{14\cdots 75}a^{17}-\frac{72\cdots 03}{14\cdots 75}a^{16}-\frac{78\cdots 01}{14\cdots 75}a^{15}+\frac{29\cdots 44}{28\cdots 55}a^{14}+\frac{33\cdots 41}{28\cdots 55}a^{13}+\frac{30\cdots 73}{14\cdots 75}a^{12}-\frac{24\cdots 01}{20\cdots 25}a^{11}-\frac{31\cdots 93}{14\cdots 75}a^{10}+\frac{56\cdots 93}{14\cdots 75}a^{9}+\frac{26\cdots 47}{14\cdots 75}a^{8}+\frac{53\cdots 89}{28\cdots 55}a^{7}-\frac{37\cdots 99}{14\cdots 75}a^{6}-\frac{14\cdots 73}{14\cdots 75}a^{5}-\frac{20\cdots 01}{14\cdots 75}a^{4}-\frac{67\cdots 00}{56\cdots 91}a^{3}-\frac{91\cdots 51}{14\cdots 75}a^{2}-\frac{62\cdots 01}{28\cdots 55}a-\frac{25\cdots 81}{56\cdots 91}$, $\frac{27\cdots 08}{20\cdots 25}a^{17}-\frac{50\cdots 87}{42\cdots 25}a^{16}-\frac{14\cdots 39}{42\cdots 25}a^{15}+\frac{71\cdots 68}{16\cdots 73}a^{14}+\frac{22\cdots 73}{28\cdots 55}a^{13}-\frac{39\cdots 39}{60\cdots 75}a^{12}-\frac{68\cdots 03}{42\cdots 25}a^{11}+\frac{20\cdots 83}{42\cdots 25}a^{10}+\frac{38\cdots 27}{20\cdots 25}a^{9}-\frac{19\cdots 82}{42\cdots 25}a^{8}-\frac{87\cdots 22}{84\cdots 65}a^{7}-\frac{58\cdots 01}{42\cdots 25}a^{6}+\frac{63\cdots 79}{60\cdots 75}a^{5}+\frac{21\cdots 86}{42\cdots 25}a^{4}+\frac{53\cdots 34}{84\cdots 65}a^{3}+\frac{17\cdots 26}{42\cdots 25}a^{2}+\frac{96\cdots 74}{56\cdots 91}a+\frac{67\cdots 83}{16\cdots 73}$ (assuming GRH)
Regulator:		$625949725971$ (assuming GRH)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{7}\cdot 625949725971 \cdot 1}{2\cdot\sqrt{14616700658977641856215848806693687375}}\cr\approx \mathstrut & 0.506365190150983 \end{aligned}$ (assuming GRH)

Galois group

$C_3^6.C_2\wr C_6$ (as 18T858):

A solvable group of order 279936

The 159 conjugacy class representatives for $C_3^6.C_2\wr C_6$

Character table for $C_3^6.C_2\wr C_6$

Intermediate fields

$\Q(\zeta_{7})^+$, 6.4.1006019.1

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

Sibling fields

Degree 18 sibling:	data not computed
Degree 24 sibling:	data not computed
Degree 27 sibling:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	18.8.3838671405368745874099609375.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$	${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{4}$	R	R	$18$	${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$	${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$	$18$	${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$	${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$	$18$	${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$	R	${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$	${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$	${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$

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
$5$	5.3.1.0a1.1	$x^{3} + 3 x + 3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	5.3.2.3a1.2	$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.9.1.0a1.1	$x^{9} + 2 x^{3} + x + 3$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$
$7$	7.1.3.2a1.1	$x^{3} + 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.1.3.2a1.1	$x^{3} + 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.1.3.2a1.1	$x^{3} + 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.1.3.2a1.1	$x^{3} + 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.2.3.4a1.2	$x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
$41$	41.2.1.0a1.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.1.0a1.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.1.0a1.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.3.1.0a1.1	$x^{3} + x + 35$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	41.3.1.0a1.1	$x^{3} + x + 35$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	41.2.3.4a1.1	$x^{6} + 114 x^{5} + 4350 x^{4} + 56240 x^{3} + 26100 x^{2} + 4145 x + 216$	$3$	$2$	$4$	$S_3\times C_3$	$[\ ]_{3}^{6}$
$419$		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
		Deg $6$	$2$	$3$	$3$
$449$		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
		Deg $6$	$3$	$2$	$4$

Spectrum of ring of integers