Number field 31.31.606473279060866185803408353837168008326030742006308575695635765007438922801.1

• Label: 31.31.606...801.1
• Degree: $31$
• Signature: $[31, 0]$
• Discriminant: $6.065\times 10^{74}$
• Root discriminant: \(258.43\)
• Ramified prime: $311$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{31}$ (as 31T1)

Normalized defining polynomial

x^31 - x^30 - 150*x^29 + 117*x^28 + 9434*x^27 - 4958*x^26 - 328880*x^25 + 78328*x^24 + 7098810*x^23 + 507925*x^22 - 100297691*x^21 - 39217806*x^20 + 950942678*x^19 + 677567200*x^18 - 6056859921*x^17 - 6293114670*x^16 + 25303767350*x^15 + 35315191500*x^14 - 65145555320*x^13 - 122001770825*x^12 + 87706314807*x^11 + 253609179978*x^10 - 19776440614*x^9 - 299693882286*x^8 - 98200795275*x^7 + 175359843704*x^6 + 113820457794*x^5 - 30260338943*x^4 - 38729335872*x^3 - 5811762482*x^2 + 1056847214*x - 26458109

Invariants

Degree:		$31$
Signature:		$[31, 0]$
Discriminant:		\(606473279060866185803408353837168008326030742006308575695635765007438922801\) \(\medspace = 311^{30}\)
Root discriminant:		\(258.43\)
Galois root discriminant:		$311^{30/31}\approx 258.43347277162707$
Ramified primes:		\(311\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$31$
This field is Galois and abelian over $\Q$.
Conductor:		\(311\)
Dirichlet character group:		$\lbrace$$\chi_{311}(1,·)$, $\chi_{311}(224,·)$, $\chi_{311}(195,·)$, $\chi_{311}(260,·)$, $\chi_{311}(7,·)$, $\chi_{311}(265,·)$, $\chi_{311}(140,·)$, $\chi_{311}(13,·)$, $\chi_{311}(270,·)$, $\chi_{311}(15,·)$, $\chi_{311}(18,·)$, $\chi_{311}(83,·)$, $\chi_{311}(20,·)$, $\chi_{311}(24,·)$, $\chi_{311}(89,·)$, $\chi_{311}(91,·)$, $\chi_{311}(32,·)$, $\chi_{311}(225,·)$, $\chi_{311}(113,·)$, $\chi_{311}(168,·)$, $\chi_{311}(169,·)$, $\chi_{311}(234,·)$, $\chi_{311}(300,·)$, $\chi_{311}(146,·)$, $\chi_{311}(47,·)$, $\chi_{311}(49,·)$, $\chi_{311}(243,·)$, $\chi_{311}(105,·)$, $\chi_{311}(121,·)$, $\chi_{311}(250,·)$, $\chi_{311}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{347}a^{28}-\frac{34}{347}a^{27}-\frac{55}{347}a^{26}+\frac{56}{347}a^{25}-\frac{63}{347}a^{24}-\frac{47}{347}a^{23}+\frac{73}{347}a^{22}+\frac{24}{347}a^{21}-\frac{16}{347}a^{20}-\frac{95}{347}a^{19}+\frac{77}{347}a^{18}-\frac{75}{347}a^{17}+\frac{117}{347}a^{16}+\frac{143}{347}a^{15}-\frac{4}{347}a^{14}-\frac{15}{347}a^{13}+\frac{21}{347}a^{12}-\frac{103}{347}a^{11}-\frac{120}{347}a^{10}-\frac{42}{347}a^{9}+\frac{58}{347}a^{8}+\frac{27}{347}a^{7}-\frac{8}{347}a^{6}+\frac{117}{347}a^{5}-\frac{138}{347}a^{4}+\frac{86}{347}a^{3}-\frac{43}{347}a^{2}-\frac{139}{347}a-\frac{149}{347}$, $\frac{1}{347}a^{29}-\frac{170}{347}a^{27}-\frac{79}{347}a^{26}+\frac{106}{347}a^{25}-\frac{107}{347}a^{24}-\frac{137}{347}a^{23}+\frac{77}{347}a^{22}+\frac{106}{347}a^{21}+\frac{55}{347}a^{20}-\frac{30}{347}a^{19}+\frac{114}{347}a^{18}-\frac{4}{347}a^{17}-\frac{43}{347}a^{16}-\frac{151}{347}a^{14}-\frac{142}{347}a^{13}-\frac{83}{347}a^{12}-\frac{152}{347}a^{11}+\frac{42}{347}a^{10}+\frac{18}{347}a^{9}-\frac{83}{347}a^{8}-\frac{131}{347}a^{7}-\frac{155}{347}a^{6}+\frac{23}{347}a^{5}-\frac{95}{347}a^{4}+\frac{105}{347}a^{3}+\frac{134}{347}a^{2}-\frac{17}{347}a+\frac{139}{347}$, $\frac{1}{54\!\cdots\!77}a^{30}+\frac{72\!\cdots\!30}{54\!\cdots\!77}a^{29}-\frac{45\!\cdots\!57}{54\!\cdots\!77}a^{28}-\frac{25\!\cdots\!86}{54\!\cdots\!77}a^{27}-\frac{46\!\cdots\!70}{54\!\cdots\!77}a^{26}-\frac{41\!\cdots\!26}{54\!\cdots\!77}a^{25}+\frac{52\!\cdots\!13}{15\!\cdots\!91}a^{24}-\frac{12\!\cdots\!30}{54\!\cdots\!77}a^{23}-\frac{34\!\cdots\!31}{54\!\cdots\!77}a^{22}+\frac{16\!\cdots\!07}{54\!\cdots\!77}a^{21}+\frac{25\!\cdots\!94}{54\!\cdots\!77}a^{20}-\frac{18\!\cdots\!38}{54\!\cdots\!77}a^{19}-\frac{13\!\cdots\!55}{54\!\cdots\!77}a^{18}-\frac{20\!\cdots\!21}{54\!\cdots\!77}a^{17}-\frac{16\!\cdots\!86}{54\!\cdots\!77}a^{16}+\frac{25\!\cdots\!34}{54\!\cdots\!77}a^{15}-\frac{24\!\cdots\!12}{54\!\cdots\!77}a^{14}-\frac{27\!\cdots\!82}{54\!\cdots\!77}a^{13}-\frac{20\!\cdots\!59}{54\!\cdots\!77}a^{12}-\frac{99\!\cdots\!31}{54\!\cdots\!77}a^{11}-\frac{13\!\cdots\!97}{54\!\cdots\!77}a^{10}+\frac{49\!\cdots\!43}{54\!\cdots\!77}a^{9}-\frac{32\!\cdots\!69}{54\!\cdots\!77}a^{8}+\frac{17\!\cdots\!13}{54\!\cdots\!77}a^{7}+\frac{21\!\cdots\!98}{54\!\cdots\!77}a^{6}+\frac{53\!\cdots\!52}{54\!\cdots\!77}a^{5}+\frac{14\!\cdots\!77}{54\!\cdots\!77}a^{4}-\frac{35\!\cdots\!45}{54\!\cdots\!77}a^{3}-\frac{17\!\cdots\!12}{54\!\cdots\!77}a^{2}-\frac{14\!\cdots\!62}{54\!\cdots\!77}a-\frac{77\!\cdots\!90}{54\!\cdots\!77}$

Monogenic:		No
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$30$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

Galois group

$C_{31}$ (as 31T1):

A cyclic group of order 31

The 31 conjugacy class representatives for $C_{31}$

Character table for $C_{31}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$	$31$

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
\(311\)		Deg $31$	$31$	$1$	$30$