Global number field 30.30.69503752297329754905479727341904896738456941915804813.1

• Label: 30.30.6950375229...4813.1
• Degree: $30$
• Signature: $[30, 0]$
• Discriminant: $11^{24}\cdot 13^{25}$
• Root discriminant: $57.73$
• Ramified primes: $11, 13$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{30}$ (as 30T1)

Normalized defining polynomial

\( x^{30} - x^{29} - 49 x^{28} + 44 x^{27} + 1012 x^{26} - 813 x^{25} - 11582 x^{24} + 8267 x^{23} + 81301 x^{22} - 51172 x^{21} - 366235 x^{20} + 201754 x^{19} + 1077688 x^{18} - 515602 x^{17} - 2071990 x^{16} + 853441 x^{15} + 2566790 x^{14} - 898459 x^{13} - 1995511 x^{12} + 578875 x^{11} + 941075 x^{10} - 215958 x^{9} - 261009 x^{8} + 45664 x^{7} + 40851 x^{6} - 5360 x^{5} - 3328 x^{4} + 330 x^{3} + 117 x^{2} - 9 x - 1 \)

Invariants

Degree:		$30$
Signature:		$[30, 0]$
Discriminant:		\(69503752297329754905479727341904896738456941915804813=11^{24}\cdot 13^{25}\)
Root discriminant:		$57.73$
Ramified primes:		$11, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(143=11\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{143}(64,·)$, $\chi_{143}(1,·)$, $\chi_{143}(3,·)$, $\chi_{143}(4,·)$, $\chi_{143}(133,·)$, $\chi_{143}(100,·)$, $\chi_{143}(113,·)$, $\chi_{143}(9,·)$, $\chi_{143}(75,·)$, $\chi_{143}(12,·)$, $\chi_{143}(14,·)$, $\chi_{143}(16,·)$, $\chi_{143}(81,·)$, $\chi_{143}(82,·)$, $\chi_{143}(23,·)$, $\chi_{143}(25,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(69,·)$, $\chi_{143}(36,·)$, $\chi_{143}(38,·)$, $\chi_{143}(103,·)$, $\chi_{143}(42,·)$, $\chi_{143}(108,·)$, $\chi_{143}(48,·)$, $\chi_{143}(49,·)$, $\chi_{143}(114,·)$, $\chi_{143}(53,·)$, $\chi_{143}(56,·)$, $\chi_{143}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{131} a^{27} + \frac{43}{131} a^{26} + \frac{15}{131} a^{25} - \frac{58}{131} a^{24} + \frac{16}{131} a^{23} - \frac{37}{131} a^{22} + \frac{65}{131} a^{21} - \frac{44}{131} a^{20} - \frac{12}{131} a^{19} + \frac{29}{131} a^{18} + \frac{13}{131} a^{17} + \frac{31}{131} a^{16} - \frac{22}{131} a^{15} - \frac{11}{131} a^{14} + \frac{27}{131} a^{13} - \frac{42}{131} a^{12} - \frac{53}{131} a^{11} - \frac{55}{131} a^{10} + \frac{28}{131} a^{9} + \frac{59}{131} a^{8} + \frac{16}{131} a^{7} - \frac{62}{131} a^{6} + \frac{11}{131} a^{5} - \frac{19}{131} a^{4} - \frac{38}{131} a^{3} - \frac{26}{131} a^{2} + \frac{8}{131} a - \frac{14}{131}$, $\frac{1}{131} a^{28} - \frac{48}{131} a^{25} + \frac{21}{131} a^{24} + \frac{61}{131} a^{23} - \frac{47}{131} a^{22} + \frac{43}{131} a^{21} + \frac{46}{131} a^{20} + \frac{21}{131} a^{19} - \frac{55}{131} a^{18} - \frac{4}{131} a^{17} - \frac{45}{131} a^{16} + \frac{18}{131} a^{15} - \frac{24}{131} a^{14} - \frac{24}{131} a^{13} + \frac{50}{131} a^{12} - \frac{3}{131} a^{11} + \frac{35}{131} a^{10} + \frac{34}{131} a^{9} - \frac{32}{131} a^{8} + \frac{36}{131} a^{7} + \frac{57}{131} a^{6} + \frac{32}{131} a^{5} - \frac{7}{131} a^{4} + \frac{36}{131} a^{3} - \frac{53}{131} a^{2} + \frac{35}{131} a - \frac{53}{131}$, $\frac{1}{616537124762503966057151703873365739834955781} a^{29} - \frac{1858876207217761109478379885851256717890099}{616537124762503966057151703873365739834955781} a^{28} - \frac{396328004214736982931294455040567183419262}{616537124762503966057151703873365739834955781} a^{27} + \frac{246276653871961532786865765679233450009974893}{616537124762503966057151703873365739834955781} a^{26} + \frac{76247795954495324461127377451972851214983127}{616537124762503966057151703873365739834955781} a^{25} - \frac{58287849949403776555234631311609091406933462}{616537124762503966057151703873365739834955781} a^{24} + \frac{178455913268925159344697090872472272231576255}{616537124762503966057151703873365739834955781} a^{23} + \frac{233220437968818537962935008822115850533095692}{616537124762503966057151703873365739834955781} a^{22} - \frac{195098778723701555418775385323934047412177334}{616537124762503966057151703873365739834955781} a^{21} - \frac{89389161648687044057597516698823714515941720}{616537124762503966057151703873365739834955781} a^{20} - \frac{53193969804121388689175647365191986692032658}{616537124762503966057151703873365739834955781} a^{19} + \frac{138665119284342606633910889893066681784358285}{616537124762503966057151703873365739834955781} a^{18} - \frac{171496994209386905911711361903633589259674127}{616537124762503966057151703873365739834955781} a^{17} - \frac{135116913904443593533661796373311521371147420}{616537124762503966057151703873365739834955781} a^{16} - \frac{144234555665866566962510576880735243741092182}{616537124762503966057151703873365739834955781} a^{15} - \frac{306880255410148637013306040673702466537464531}{616537124762503966057151703873365739834955781} a^{14} + \frac{40915825328516183937689530788350816250535830}{616537124762503966057151703873365739834955781} a^{13} + \frac{139896590913101583827125339093851200002846583}{616537124762503966057151703873365739834955781} a^{12} + \frac{40983396800866632184453051167450953430922941}{616537124762503966057151703873365739834955781} a^{11} - \frac{241887803547477730130153017437999112130551138}{616537124762503966057151703873365739834955781} a^{10} + \frac{300022408162222712506325958389721753091024579}{616537124762503966057151703873365739834955781} a^{9} + \frac{148473882914844606484970803020957488686473557}{616537124762503966057151703873365739834955781} a^{8} - \frac{62537130476008394396668297893288119395927420}{616537124762503966057151703873365739834955781} a^{7} + \frac{153195924487865642959704615179958590800442150}{616537124762503966057151703873365739834955781} a^{6} + \frac{137786901201764042287401600512235060893935817}{616537124762503966057151703873365739834955781} a^{5} - \frac{108403151855840829708140204081537554757942427}{616537124762503966057151703873365739834955781} a^{4} - \frac{467104045434569929160290293264703168375194}{616537124762503966057151703873365739834955781} a^{3} - \frac{41761681818335908650690084082325462657316442}{616537124762503966057151703873365739834955781} a^{2} - \frac{5721714962908458545332027262085324777182201}{616537124762503966057151703873365739834955781} a + \frac{156146195994799579076472615871903214561517901}{616537124762503966057151703873365739834955781}$

Class group and class number

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

Unit group

Rank:		$29$
Torsion generator:		\( -1 \) (order $2$)
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:		\( 59204301251900104 \) (assuming GRH)

Galois group

$C_{30}$ (as 30T1):

A cyclic group of order 30

The 30 conjugacy class representatives for $C_{30}$

Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{13})^+\), 10.10.79589952003133.1, 15.15.432659002790862279847129.1

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

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$30$	$15^{2}$	${\href{/LocalNumberField/5.10.0.1}{10} }^{3}$	$30$	R	R	$15^{2}$	$30$	${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$	$15^{2}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{3}$	$30$	$30$	${\href{/LocalNumberField/43.3.0.1}{3} }^{10}$	${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$	$30$

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
11	Data not computed
13	Data not computed