Number field 43.43.444677695956607074780919035502815976195331208356891344496189409566167816104341684988715963441514628066825276961.1

• Label: 43.43.444...961.1
• Degree: $43$
• Signature: $(43, 0)$
• Discriminant: $4.447\times 10^{110}$
• Root discriminant: \(374.29\)
• Ramified prime: $431$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{43}$ (as 43T1)

Normalized defining polynomial

x^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501

Invariants

Degree:		$43$
Signature:		$(43, 0)$
Discriminant:		444677695956607074780919035502815976195331208356891344496189409566167816104341684988715963441514628066825276961
Root discriminant:		\(374.29\)
Galois root discriminant:		$431^{42/43}\approx 374.29183811372997$
Ramified primes:		\(431\)
Discriminant root field:		\(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:		$C_{43}$
This field is Galois and abelian over $\Q$.
Conductor:		\(431\)
Dirichlet character group:		$\lbrace$$\chi_{431}(128,·)$, $\chi_{431}(1,·)$, $\chi_{431}(2,·)$, $\chi_{431}(3,·)$, $\chi_{431}(4,·)$, $\chi_{431}(6,·)$, $\chi_{431}(8,·)$, $\chi_{431}(9,·)$, $\chi_{431}(12,·)$, $\chi_{431}(256,·)$, $\chi_{431}(16,·)$, $\chi_{431}(145,·)$, $\chi_{431}(18,·)$, $\chi_{431}(149,·)$, $\chi_{431}(24,·)$, $\chi_{431}(27,·)$, $\chi_{431}(32,·)$, $\chi_{431}(162,·)$, $\chi_{431}(36,·)$, $\chi_{431}(165,·)$, $\chi_{431}(64,·)$, $\chi_{431}(298,·)$, $\chi_{431}(48,·)$, $\chi_{431}(54,·)$, $\chi_{431}(55,·)$, $\chi_{431}(192,·)$, $\chi_{431}(288,·)$, $\chi_{431}(324,·)$, $\chi_{431}(72,·)$, $\chi_{431}(330,·)$, $\chi_{431}(290,·)$, $\chi_{431}(81,·)$, $\chi_{431}(216,·)$, $\chi_{431}(217,·)$, $\chi_{431}(384,·)$, $\chi_{431}(220,·)$, $\chi_{431}(96,·)$, $\chi_{431}(144,·)$, $\chi_{431}(229,·)$, $\chi_{431}(337,·)$, $\chi_{431}(108,·)$, $\chi_{431}(110,·)$, $\chi_{431}(243,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{617}a^{40}-\frac{96}{617}a^{39}-\frac{38}{617}a^{38}-\frac{212}{617}a^{37}-\frac{104}{617}a^{36}-\frac{244}{617}a^{35}+\frac{238}{617}a^{34}-\frac{176}{617}a^{33}-\frac{13}{617}a^{32}+\frac{269}{617}a^{31}-\frac{48}{617}a^{30}-\frac{85}{617}a^{29}+\frac{101}{617}a^{28}-\frac{109}{617}a^{27}-\frac{293}{617}a^{26}-\frac{87}{617}a^{25}-\frac{192}{617}a^{24}-\frac{36}{617}a^{23}-\frac{264}{617}a^{22}-\frac{84}{617}a^{21}-\frac{289}{617}a^{20}+\frac{224}{617}a^{19}+\frac{114}{617}a^{18}+\frac{236}{617}a^{17}-\frac{43}{617}a^{16}+\frac{293}{617}a^{15}+\frac{198}{617}a^{14}-\frac{239}{617}a^{13}+\frac{243}{617}a^{12}-\frac{265}{617}a^{11}+\frac{301}{617}a^{10}-\frac{117}{617}a^{9}+\frac{281}{617}a^{8}-\frac{235}{617}a^{7}+\frac{215}{617}a^{6}+\frac{308}{617}a^{5}-\frac{83}{617}a^{4}-\frac{166}{617}a^{3}+\frac{245}{617}a^{2}-\frac{51}{617}a+\frac{248}{617}$, $\frac{1}{320964466793}a^{41}+\frac{139796710}{320964466793}a^{40}+\frac{28941309337}{320964466793}a^{39}+\frac{126390064522}{320964466793}a^{38}+\frac{66208018678}{320964466793}a^{37}+\frac{14066083648}{320964466793}a^{36}+\frac{87870452426}{320964466793}a^{35}-\frac{15882023825}{320964466793}a^{34}-\frac{117381994781}{320964466793}a^{33}+\frac{51354677571}{320964466793}a^{32}+\frac{31659308681}{320964466793}a^{31}+\frac{26506178368}{320964466793}a^{30}+\frac{72390307907}{320964466793}a^{29}-\frac{156929706337}{320964466793}a^{28}+\frac{30802602707}{320964466793}a^{27}+\frac{81085137460}{320964466793}a^{26}+\frac{158840472175}{320964466793}a^{25}-\frac{153999837168}{320964466793}a^{24}+\frac{131668817904}{320964466793}a^{23}-\frac{123034361111}{320964466793}a^{22}-\frac{85402152911}{320964466793}a^{21}-\frac{117152204418}{320964466793}a^{20}+\frac{117058398997}{320964466793}a^{19}+\frac{90109407825}{320964466793}a^{18}+\frac{145433894181}{320964466793}a^{17}-\frac{34944252620}{320964466793}a^{16}+\frac{95471731337}{320964466793}a^{15}-\frac{77725354228}{320964466793}a^{14}-\frac{99566354704}{320964466793}a^{13}+\frac{41397120030}{320964466793}a^{12}+\frac{119638790936}{320964466793}a^{11}+\frac{55272436394}{320964466793}a^{10}-\frac{77662797205}{320964466793}a^{9}+\frac{58406679660}{320964466793}a^{8}+\frac{88370551543}{320964466793}a^{7}-\frac{61647862697}{320964466793}a^{6}-\frac{33155475203}{320964466793}a^{5}-\frac{33723425418}{320964466793}a^{4}+\frac{93792925557}{320964466793}a^{3}-\frac{97241821493}{320964466793}a^{2}+\frac{23328690925}{320964466793}a-\frac{107274973111}{320964466793}$, $\frac{1}{14\cdots 37}a^{42}-\frac{22\cdots 44}{14\cdots 37}a^{41}+\frac{11\cdots 72}{14\cdots 37}a^{40}-\frac{66\cdots 58}{14\cdots 37}a^{39}+\frac{25\cdots 94}{14\cdots 37}a^{38}-\frac{61\cdots 28}{14\cdots 37}a^{37}+\frac{65\cdots 97}{14\cdots 37}a^{36}+\frac{44\cdots 45}{14\cdots 37}a^{35}-\frac{41\cdots 24}{14\cdots 37}a^{34}-\frac{14\cdots 33}{14\cdots 37}a^{33}-\frac{47\cdots 66}{14\cdots 37}a^{32}-\frac{17\cdots 80}{14\cdots 37}a^{31}-\frac{64\cdots 37}{14\cdots 37}a^{30}+\frac{62\cdots 71}{14\cdots 37}a^{29}+\frac{29\cdots 34}{14\cdots 37}a^{28}-\frac{19\cdots 95}{14\cdots 37}a^{27}+\frac{95\cdots 04}{14\cdots 37}a^{26}+\frac{10\cdots 42}{14\cdots 37}a^{25}+\frac{66\cdots 21}{14\cdots 37}a^{24}+\frac{66\cdots 88}{14\cdots 37}a^{23}-\frac{52\cdots 52}{14\cdots 37}a^{22}+\frac{49\cdots 09}{14\cdots 37}a^{21}+\frac{41\cdots 80}{14\cdots 37}a^{20}+\frac{48\cdots 27}{14\cdots 37}a^{19}-\frac{25\cdots 98}{14\cdots 37}a^{18}+\frac{55\cdots 45}{14\cdots 37}a^{17}+\frac{62\cdots 99}{14\cdots 37}a^{16}+\frac{57\cdots 15}{14\cdots 37}a^{15}-\frac{60\cdots 54}{14\cdots 37}a^{14}-\frac{71\cdots 58}{14\cdots 37}a^{13}-\frac{77\cdots 19}{23\cdots 61}a^{12}-\frac{42\cdots 38}{14\cdots 37}a^{11}+\frac{33\cdots 80}{14\cdots 37}a^{10}+\frac{97\cdots 15}{23\cdots 61}a^{9}+\frac{21\cdots 41}{14\cdots 37}a^{8}-\frac{37\cdots 20}{14\cdots 37}a^{7}-\frac{44\cdots 01}{14\cdots 37}a^{6}+\frac{19\cdots 19}{14\cdots 37}a^{5}+\frac{26\cdots 23}{14\cdots 37}a^{4}+\frac{26\cdots 89}{14\cdots 37}a^{3}-\frac{36\cdots 53}{14\cdots 37}a^{2}+\frac{16\cdots 33}{14\cdots 37}a-\frac{22\cdots 52}{45\cdots 79}$

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

Class group and class number

Ideal class group:		not computed
Narrow class group:		not computed

Unit group

Rank:		$42$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed
Unit signature rank:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{43}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{444677695956607074780919035502815976195331208356891344496189409566167816104341684988715963441514628066825276961}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Galois group

$C_{43}$ (as 43T1):

A cyclic group of order 43

The 43 conjugacy class representatives for $C_{43}$

Character table for $C_{43}$

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	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$	$43$

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
\(431\)		Deg $43$	$43$	$1$	$42$

Spectrum of ring of integers