Number field 35.35.31388512296654191827836498790055109793463688586358814341907093630301198422556230514776488761.1

• Label: 35.35.313...761.1
• Degree: $35$
• Signature: $[35, 0]$
• Discriminant: $3.139\times 10^{91}$
• Root discriminant: \(411.33\)
• Ramified prime: $491$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{35}$ (as 35T1)

Normalized defining polynomial

x^35 - x^34 - 238*x^33 + 173*x^32 + 24286*x^31 - 16434*x^30 - 1418466*x^29 + 1081623*x^28 + 53047154*x^27 - 49719432*x^26 - 1339060322*x^25 + 1572006663*x^24 + 23323700077*x^23 - 34062526929*x^22 - 280163543133*x^21 + 504952475173*x^20 + 2262920124956*x^19 - 5071803868827*x^18 - 11476437360382*x^17 + 33655878701114*x^16 + 29727135340615*x^15 - 140118838407360*x^14 + 3247425380495*x^13 + 329245748251530*x^12 - 225104736115922*x^11 - 332674574789057*x^10 + 467468058300787*x^9 - 15355505673450*x^8 - 263968105366926*x^7 + 131023125506686*x^6 + 23123425964865*x^5 - 32591926570801*x^4 + 5190946345187*x^3 + 1696655472839*x^2 - 515233369184*x + 22178194211

Invariants

Degree:		$35$
Signature:		$[35, 0]$
Discriminant:		31388512296654191827836498790055109793463688586358814341907093630301198422556230514776488761
Root discriminant:		\(411.33\)
Galois root discriminant:		$491^{34/35}\approx 411.33290192846266$
Ramified primes:		\(491\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$35$
This field is Galois and abelian over $\Q$.
Conductor:		\(491\)
Dirichlet character group:		$\lbrace$$\chi_{491}(1,·)$, $\chi_{491}(386,·)$, $\chi_{491}(257,·)$, $\chi_{491}(138,·)$, $\chi_{491}(12,·)$, $\chi_{491}(144,·)$, $\chi_{491}(20,·)$, $\chi_{491}(153,·)$, $\chi_{491}(41,·)$, $\chi_{491}(428,·)$, $\chi_{491}(305,·)$, $\chi_{491}(181,·)$, $\chi_{491}(183,·)$, $\chi_{491}(56,·)$, $\chi_{491}(316,·)$, $\chi_{491}(190,·)$, $\chi_{491}(197,·)$, $\chi_{491}(329,·)$, $\chi_{491}(332,·)$, $\chi_{491}(208,·)$, $\chi_{491}(213,·)$, $\chi_{491}(221,·)$, $\chi_{491}(223,·)$, $\chi_{491}(400,·)$, $\chi_{491}(226,·)$, $\chi_{491}(355,·)$, $\chi_{491}(101,·)$, $\chi_{491}(230,·)$, $\chi_{491}(232,·)$, $\chi_{491}(363,·)$, $\chi_{491}(240,·)$, $\chi_{491}(114,·)$, $\chi_{491}(425,·)$, $\chi_{491}(381,·)$, $\chi_{491}(255,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{55973}a^{33}+\frac{14099}{55973}a^{32}-\frac{9356}{55973}a^{31}-\frac{22987}{55973}a^{30}+\frac{7484}{55973}a^{29}+\frac{26025}{55973}a^{28}-\frac{13890}{55973}a^{27}+\frac{30}{251}a^{26}-\frac{4398}{55973}a^{25}-\frac{5556}{55973}a^{24}-\frac{20177}{55973}a^{23}-\frac{15024}{55973}a^{22}-\frac{26343}{55973}a^{21}-\frac{17475}{55973}a^{20}+\frac{27963}{55973}a^{19}-\frac{25089}{55973}a^{18}-\frac{19965}{55973}a^{17}+\frac{9288}{55973}a^{16}-\frac{22775}{55973}a^{15}-\frac{22188}{55973}a^{14}-\frac{20518}{55973}a^{13}+\frac{18977}{55973}a^{12}+\frac{19428}{55973}a^{11}+\frac{2804}{55973}a^{10}+\frac{4643}{55973}a^{9}-\frac{14875}{55973}a^{8}+\frac{20177}{55973}a^{7}+\frac{16070}{55973}a^{6}+\frac{21391}{55973}a^{5}-\frac{750}{55973}a^{4}+\frac{9132}{55973}a^{3}+\frac{19366}{55973}a^{2}+\frac{15573}{55973}a-\frac{16983}{55973}$, $\frac{1}{74\!\cdots\!61}a^{34}-\frac{55\!\cdots\!64}{74\!\cdots\!61}a^{33}+\frac{19\!\cdots\!23}{74\!\cdots\!61}a^{32}-\frac{23\!\cdots\!88}{74\!\cdots\!61}a^{31}+\frac{37\!\cdots\!08}{74\!\cdots\!61}a^{30}+\frac{18\!\cdots\!83}{74\!\cdots\!61}a^{29}+\frac{18\!\cdots\!09}{74\!\cdots\!61}a^{28}-\frac{91\!\cdots\!54}{74\!\cdots\!61}a^{27}+\frac{42\!\cdots\!11}{74\!\cdots\!61}a^{26}-\frac{29\!\cdots\!59}{74\!\cdots\!61}a^{25}-\frac{56\!\cdots\!32}{74\!\cdots\!61}a^{24}-\frac{13\!\cdots\!65}{74\!\cdots\!61}a^{23}-\frac{17\!\cdots\!27}{74\!\cdots\!61}a^{22}-\frac{23\!\cdots\!16}{74\!\cdots\!61}a^{21}+\frac{17\!\cdots\!09}{74\!\cdots\!61}a^{20}+\frac{35\!\cdots\!64}{74\!\cdots\!61}a^{19}-\frac{86\!\cdots\!22}{74\!\cdots\!61}a^{18}-\frac{24\!\cdots\!03}{74\!\cdots\!61}a^{17}-\frac{16\!\cdots\!10}{74\!\cdots\!61}a^{16}-\frac{20\!\cdots\!78}{74\!\cdots\!61}a^{15}+\frac{27\!\cdots\!91}{74\!\cdots\!61}a^{14}+\frac{33\!\cdots\!42}{74\!\cdots\!61}a^{13}-\frac{75\!\cdots\!91}{74\!\cdots\!61}a^{12}+\frac{39\!\cdots\!38}{74\!\cdots\!61}a^{11}+\frac{55\!\cdots\!89}{74\!\cdots\!61}a^{10}-\frac{16\!\cdots\!14}{74\!\cdots\!61}a^{9}+\frac{23\!\cdots\!93}{74\!\cdots\!61}a^{8}+\frac{21\!\cdots\!08}{74\!\cdots\!61}a^{7}+\frac{21\!\cdots\!08}{74\!\cdots\!61}a^{6}+\frac{36\!\cdots\!12}{74\!\cdots\!61}a^{5}+\frac{87\!\cdots\!61}{74\!\cdots\!61}a^{4}-\frac{26\!\cdots\!18}{74\!\cdots\!61}a^{3}+\frac{74\!\cdots\!84}{74\!\cdots\!61}a^{2}-\frac{12\!\cdots\!34}{74\!\cdots\!61}a+\frac{34\!\cdots\!66}{74\!\cdots\!61}$

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

Class group and class number

not computed

Unit group

Rank:		$34$
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

$C_{35}$ (as 35T1):

A cyclic group of order 35

The 35 conjugacy class representatives for $C_{35}$

Character table for $C_{35}$ is not computed

Intermediate fields

5.5.58120048561.1, 7.7.14011639427134441.1

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	$35$	${\href{/padicField/3.7.0.1}{7} }^{5}$	$35$	$35$	$35$	$35$	${\href{/padicField/17.7.0.1}{7} }^{5}$	$35$	$35$	$35$	$35$	${\href{/padicField/37.7.0.1}{7} }^{5}$	${\href{/padicField/41.5.0.1}{5} }^{7}$	${\href{/padicField/43.7.0.1}{7} }^{5}$	$35$	${\href{/padicField/53.7.0.1}{7} }^{5}$	$35$

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
\(491\)		Deg $35$	$35$	$1$	$34$