Number field 30.0.45711723244122563996456104229882472771.1

• Label: 30.0.457...771.1
• Degree: $30$
• Signature: $[0, 15]$
• Discriminant: $-4.571\times 10^{37}$
• Root discriminant: $18.00$
• Ramified primes: $3, 11$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{15}\times S_3$ (as 30T15)

Normalized defining polynomial

\( x^{30} + 6 x^{28} - x^{27} + 14 x^{26} - x^{25} + 8 x^{24} + 13 x^{23} - 39 x^{22} + 37 x^{21} - 93 x^{20} + 30 x^{19} + 10 x^{18} - 13 x^{17} + 349 x^{16} + 77 x^{15} + 721 x^{14} + 575 x^{13} + 1026 x^{12} + 1233 x^{11} + 1137 x^{10} + 993 x^{9} + 501 x^{8} - 41 x^{7} - 253 x^{6} - 186 x^{5} - 58 x^{4} + 6 x^{3} + 14 x^{2} + 6 x + 1 \)

Invariants

Degree:		$30$
Signature:		$[0, 15]$
Discriminant:		\(-45711723244122563996456104229882472771\)\(\medspace = -\,3^{20}\cdot 11^{27}\)
Root discriminant:		$18.00$
Ramified primes:		$3, 11$
$|\Aut(K/\Q)|$:		$15$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{445} a^{25} + \frac{11}{445} a^{24} - \frac{219}{445} a^{23} + \frac{86}{445} a^{22} + \frac{39}{89} a^{21} + \frac{186}{445} a^{20} + \frac{82}{445} a^{19} - \frac{26}{89} a^{18} + \frac{203}{445} a^{17} + \frac{153}{445} a^{16} + \frac{167}{445} a^{15} + \frac{169}{445} a^{14} + \frac{23}{89} a^{13} - \frac{34}{89} a^{12} + \frac{51}{445} a^{11} - \frac{167}{445} a^{10} - \frac{164}{445} a^{9} - \frac{25}{89} a^{8} - \frac{10}{89} a^{7} + \frac{84}{445} a^{6} - \frac{4}{445} a^{5} + \frac{212}{445} a^{4} + \frac{18}{89} a^{3} - \frac{32}{89} a^{2} + \frac{18}{445} a + \frac{152}{445}$, $\frac{1}{445} a^{26} + \frac{21}{89} a^{24} - \frac{35}{89} a^{23} + \frac{139}{445} a^{22} - \frac{179}{445} a^{21} - \frac{184}{445} a^{20} - \frac{142}{445} a^{19} - \frac{147}{445} a^{18} + \frac{29}{89} a^{17} - \frac{181}{445} a^{16} + \frac{112}{445} a^{15} + \frac{36}{445} a^{14} - \frac{20}{89} a^{13} + \frac{141}{445} a^{12} + \frac{162}{445} a^{11} - \frac{107}{445} a^{10} - \frac{101}{445} a^{9} - \frac{2}{89} a^{8} + \frac{189}{445} a^{7} - \frac{38}{445} a^{6} - \frac{189}{445} a^{5} - \frac{17}{445} a^{4} + \frac{37}{89} a^{3} - \frac{2}{445} a^{2} - \frac{46}{445} a + \frac{108}{445}$, $\frac{1}{445} a^{27} + \frac{1}{89} a^{24} - \frac{6}{445} a^{23} + \frac{136}{445} a^{22} - \frac{189}{445} a^{21} - \frac{92}{445} a^{20} + \frac{143}{445} a^{19} - \frac{136}{445} a^{17} + \frac{67}{445} a^{16} - \frac{144}{445} a^{15} - \frac{9}{89} a^{14} + \frac{81}{445} a^{13} + \frac{212}{445} a^{12} - \frac{122}{445} a^{11} + \frac{79}{445} a^{10} - \frac{29}{89} a^{9} - \frac{36}{445} a^{8} - \frac{128}{445} a^{7} - \frac{109}{445} a^{6} - \frac{42}{445} a^{5} + \frac{35}{89} a^{4} - \frac{107}{445} a^{3} - \frac{156}{445} a^{2} - \frac{2}{445} a + \frac{12}{89}$, $\frac{1}{13387012429555} a^{28} + \frac{11292184506}{13387012429555} a^{27} + \frac{239979340}{2677402485911} a^{26} - \frac{11621511658}{13387012429555} a^{25} - \frac{4902792104044}{13387012429555} a^{24} - \frac{6166493836393}{13387012429555} a^{23} - \frac{5758271901301}{13387012429555} a^{22} - \frac{1341906105356}{13387012429555} a^{21} - \frac{5466840494667}{13387012429555} a^{20} + \frac{6444606444167}{13387012429555} a^{19} + \frac{6175612326114}{13387012429555} a^{18} - \frac{1231175297543}{13387012429555} a^{17} + \frac{1579876048774}{13387012429555} a^{16} - \frac{896338943675}{2677402485911} a^{15} + \frac{4836948846369}{13387012429555} a^{14} + \frac{4601689343788}{13387012429555} a^{13} - \frac{511854111159}{2677402485911} a^{12} - \frac{5347110096196}{13387012429555} a^{11} - \frac{991991156498}{2677402485911} a^{10} - \frac{2241961602974}{13387012429555} a^{9} + \frac{2053307832501}{13387012429555} a^{8} - \frac{759610988112}{13387012429555} a^{7} + \frac{4602001636602}{13387012429555} a^{6} - \frac{848191643384}{2677402485911} a^{5} - \frac{606966174518}{13387012429555} a^{4} - \frac{1820764963473}{13387012429555} a^{3} + \frac{589929716967}{13387012429555} a^{2} - \frac{3254376689291}{13387012429555} a - \frac{2112461484861}{13387012429555}$, $\frac{1}{13387012429555} a^{29} + \frac{1002026522}{2677402485911} a^{27} + \frac{14689278362}{13387012429555} a^{26} + \frac{13605534833}{13387012429555} a^{25} + \frac{324983930549}{2677402485911} a^{24} + \frac{682285836191}{2677402485911} a^{23} - \frac{1164366560091}{2677402485911} a^{22} - \frac{1045981908}{2677402485911} a^{21} + \frac{4249606413601}{13387012429555} a^{20} + \frac{3656525880493}{13387012429555} a^{19} + \frac{1039894128888}{13387012429555} a^{18} - \frac{5903150631877}{13387012429555} a^{17} - \frac{326707186607}{2677402485911} a^{16} - \frac{3339284003187}{13387012429555} a^{15} + \frac{1170255526547}{2677402485911} a^{14} - \frac{6669905464812}{13387012429555} a^{13} + \frac{6663867702186}{13387012429555} a^{12} + \frac{3666619169923}{13387012429555} a^{11} - \frac{2379384253328}{13387012429555} a^{10} - \frac{4240524289601}{13387012429555} a^{9} - \frac{1682088816724}{13387012429555} a^{8} + \frac{293229730896}{13387012429555} a^{7} + \frac{451143399178}{2677402485911} a^{6} + \frac{2876129479224}{13387012429555} a^{5} + \frac{1849453151018}{13387012429555} a^{4} + \frac{604978445978}{13387012429555} a^{3} - \frac{2196819749374}{13387012429555} a^{2} + \frac{883452233921}{2677402485911} a - \frac{2116054177081}{13387012429555}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$14$
Torsion generator:		\( \frac{12653957152607}{2677402485911} a^{29} - \frac{6159480532392}{2677402485911} a^{28} + \frac{78477429740129}{2677402485911} a^{27} - \frac{50349640529454}{2677402485911} a^{26} + \frac{198772937530862}{2677402485911} a^{25} - \frac{105849092098559}{2677402485911} a^{24} + \frac{144663718380552}{2677402485911} a^{23} + \frac{102353127789123}{2677402485911} a^{22} - \frac{550735971791860}{2677402485911} a^{21} + \frac{736058773339872}{2677402485911} a^{20} - \frac{1513402505071755}{2677402485911} a^{19} + \frac{1078104915700476}{2677402485911} a^{18} - \frac{328577462792968}{2677402485911} a^{17} - \frac{76704618826301}{2677402485911} a^{16} + \frac{4489202749714251}{2677402485911} a^{15} - \frac{1215995815292423}{2677402485911} a^{14} + \frac{9556496914311311}{2677402485911} a^{13} + \frac{2768316816085662}{2677402485911} a^{12} + \frac{11274958798156982}{2677402485911} a^{11} + \frac{10235722776091104}{2677402485911} a^{10} + \frac{9077290907100770}{2677402485911} a^{9} + \frac{8052604766471538}{2677402485911} a^{8} + \frac{2342098007255521}{2677402485911} a^{7} - \frac{1717269073290881}{2677402485911} a^{6} - \frac{2251867087989700}{2677402485911} a^{5} - \frac{1126642969502881}{2677402485911} a^{4} - \frac{133995717584920}{2677402485911} a^{3} + \frac{136551788434983}{2677402485911} a^{2} + \frac{93114123118404}{2677402485911} a + \frac{22939094334598}{2677402485911} \) (order $22$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 27376181.549941503 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 27376181.549941503 \cdot 1}{22\sqrt{45711723244122563996456104229882472771}}\approx 0.172836084881715$ (assuming GRH)

Galois group

$C_{15}\times S_3$ (as 30T15):

A solvable group of order 90

The 45 conjugacy class representatives for $C_{15}\times S_3$

Character table for $C_{15}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 6.0.107811.1, \(\Q(\zeta_{11})\)

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

Sibling fields

Degree 45 sibling:

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	$30$	R	$15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{3}$	$30$	R	$30$	${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$	$30$	$15{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{3}$	$15^{2}$	$30$	${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$	$15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{3}$	$15^{2}$	$15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$3$	3.15.20.99	$x^{15} + 42 x^{14} + 27 x^{13} + 31 x^{12} + 45 x^{11} + 45 x^{10} + 41 x^{9} + 30 x^{8} + 42 x^{7} + 8 x^{6} + 75 x^{5} + 18 x^{4} + 42 x^{3} + 24 x^{2} + 51 x + 19$	$3$	$5$	$20$	$C_{15}$	$[2]^{5}$
	3.15.0.1	$x^{15} + x^{2} - x + 1$	$1$	$15$	$0$	$C_{15}$	$[\ ]^{15}$
11	Data not computed

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.11.2t1.a.a	$1$	$ 11 $	\(\Q(\sqrt{-11}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.99.6t1.a.a	$1$	$ 3^{2} \cdot 11 $	6.0.8732691.1	$C_6$ (as 6T1)	$0$	$-1$
	1.99.6t1.a.b	$1$	$ 3^{2} \cdot 11 $	6.0.8732691.1	$C_6$ (as 6T1)	$0$	$-1$
	1.9.3t1.a.a	$1$	$ 3^{2}$	\(\Q(\zeta_{9})^+\)	$C_3$ (as 3T1)	$0$	$1$
	1.9.3t1.a.b	$1$	$ 3^{2}$	\(\Q(\zeta_{9})^+\)	$C_3$ (as 3T1)	$0$	$1$
*	1.11.10t1.a.a	$1$	$ 11 $	\(\Q(\zeta_{11})\)	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.11.5t1.a.a	$1$	$ 11 $	\(\Q(\zeta_{11})^+\)	$C_5$ (as 5T1)	$0$	$1$
*	1.11.10t1.a.b	$1$	$ 11 $	\(\Q(\zeta_{11})\)	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.11.10t1.a.c	$1$	$ 11 $	\(\Q(\zeta_{11})\)	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.11.5t1.a.b	$1$	$ 11 $	\(\Q(\zeta_{11})^+\)	$C_5$ (as 5T1)	$0$	$1$
*	1.11.5t1.a.c	$1$	$ 11 $	\(\Q(\zeta_{11})^+\)	$C_5$ (as 5T1)	$0$	$1$
*	1.11.5t1.a.d	$1$	$ 11 $	\(\Q(\zeta_{11})^+\)	$C_5$ (as 5T1)	$0$	$1$
*	1.11.10t1.a.d	$1$	$ 11 $	\(\Q(\zeta_{11})\)	$C_{10}$ (as 10T1)	$0$	$-1$
	1.99.30t1.a.a	$1$	$ 3^{2} \cdot 11 $	30.0.159386923550435671074967363509984324121230045171.1	$C_{30}$ (as 30T1)	$0$	$-1$
	1.99.30t1.a.b	$1$	$ 3^{2} \cdot 11 $	30.0.159386923550435671074967363509984324121230045171.1	$C_{30}$ (as 30T1)	$0$	$-1$
	1.99.15t1.a.a	$1$	$ 3^{2} \cdot 11 $	15.15.10943023107606534329121.1	$C_{15}$ (as 15T1)	$0$	$1$
	1.99.30t1.a.c	$1$	$ 3^{2} \cdot 11 $	30.0.159386923550435671074967363509984324121230045171.1	$C_{30}$ (as 30T1)	$0$	$-1$
	1.99.30t1.a.d	$1$	$ 3^{2} \cdot 11 $	30.0.159386923550435671074967363509984324121230045171.1	$C_{30}$ (as 30T1)	$0$	$-1$
	1.99.15t1.a.b	$1$	$ 3^{2} \cdot 11 $	15.15.10943023107606534329121.1	$C_{15}$ (as 15T1)	$0$	$1$
	1.99.30t1.a.e	$1$	$ 3^{2} \cdot 11 $	30.0.159386923550435671074967363509984324121230045171.1	$C_{30}$ (as 30T1)	$0$	$-1$
	1.99.15t1.a.c	$1$	$ 3^{2} \cdot 11 $	15.15.10943023107606534329121.1	$C_{15}$ (as 15T1)	$0$	$1$
	1.99.30t1.a.f	$1$	$ 3^{2} \cdot 11 $	30.0.159386923550435671074967363509984324121230045171.1	$C_{30}$ (as 30T1)	$0$	$-1$
	1.99.15t1.a.d	$1$	$ 3^{2} \cdot 11 $	15.15.10943023107606534329121.1	$C_{15}$ (as 15T1)	$0$	$1$
	1.99.30t1.a.g	$1$	$ 3^{2} \cdot 11 $	30.0.159386923550435671074967363509984324121230045171.1	$C_{30}$ (as 30T1)	$0$	$-1$
	1.99.15t1.a.e	$1$	$ 3^{2} \cdot 11 $	15.15.10943023107606534329121.1	$C_{15}$ (as 15T1)	$0$	$1$
	1.99.30t1.a.h	$1$	$ 3^{2} \cdot 11 $	30.0.159386923550435671074967363509984324121230045171.1	$C_{30}$ (as 30T1)	$0$	$-1$
	1.99.15t1.a.f	$1$	$ 3^{2} \cdot 11 $	15.15.10943023107606534329121.1	$C_{15}$ (as 15T1)	$0$	$1$
	1.99.15t1.a.g	$1$	$ 3^{2} \cdot 11 $	15.15.10943023107606534329121.1	$C_{15}$ (as 15T1)	$0$	$1$
	1.99.15t1.a.h	$1$	$ 3^{2} \cdot 11 $	15.15.10943023107606534329121.1	$C_{15}$ (as 15T1)	$0$	$1$
	2.891.3t2.b.a	$2$	$ 3^{4} \cdot 11 $	3.1.891.1	$S_3$ (as 3T2)	$1$	$0$
*	2.99.6t5.a.a	$2$	$ 3^{2} \cdot 11 $	6.0.107811.1	$S_3\times C_3$ (as 6T5)	$0$	$0$
*	2.99.6t5.a.b	$2$	$ 3^{2} \cdot 11 $	6.0.107811.1	$S_3\times C_3$ (as 6T5)	$0$	$0$
	2.9801.15t4.a.a	$2$	$ 3^{4} \cdot 11^{2}$	15.5.120373254183671877620331.1	$S_3 \times C_5$ (as 15T4)	$0$	$0$
	2.9801.15t4.a.b	$2$	$ 3^{4} \cdot 11^{2}$	15.5.120373254183671877620331.1	$S_3 \times C_5$ (as 15T4)	$0$	$0$
	2.9801.15t4.a.c	$2$	$ 3^{4} \cdot 11^{2}$	15.5.120373254183671877620331.1	$S_3 \times C_5$ (as 15T4)	$0$	$0$
	2.9801.15t4.a.d	$2$	$ 3^{4} \cdot 11^{2}$	15.5.120373254183671877620331.1	$S_3 \times C_5$ (as 15T4)	$0$	$0$
*	2.1089.30t15.a.a	$2$	$ 3^{2} \cdot 11^{2}$	30.0.45711723244122563996456104229882472771.1	$C_{15}\times S_3$ (as 30T15)	$0$	$0$
*	2.1089.30t15.a.b	$2$	$ 3^{2} \cdot 11^{2}$	30.0.45711723244122563996456104229882472771.1	$C_{15}\times S_3$ (as 30T15)	$0$	$0$
*	2.1089.30t15.a.c	$2$	$ 3^{2} \cdot 11^{2}$	30.0.45711723244122563996456104229882472771.1	$C_{15}\times S_3$ (as 30T15)	$0$	$0$
*	2.1089.30t15.a.d	$2$	$ 3^{2} \cdot 11^{2}$	30.0.45711723244122563996456104229882472771.1	$C_{15}\times S_3$ (as 30T15)	$0$	$0$
*	2.1089.30t15.a.e	$2$	$ 3^{2} \cdot 11^{2}$	30.0.45711723244122563996456104229882472771.1	$C_{15}\times S_3$ (as 30T15)	$0$	$0$
*	2.1089.30t15.a.f	$2$	$ 3^{2} \cdot 11^{2}$	30.0.45711723244122563996456104229882472771.1	$C_{15}\times S_3$ (as 30T15)	$0$	$0$
*	2.1089.30t15.a.g	$2$	$ 3^{2} \cdot 11^{2}$	30.0.45711723244122563996456104229882472771.1	$C_{15}\times S_3$ (as 30T15)	$0$	$0$
*	2.1089.30t15.a.h	$2$	$ 3^{2} \cdot 11^{2}$	30.0.45711723244122563996456104229882472771.1	$C_{15}\times S_3$ (as 30T15)	$0$	$0$

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 *.