Number field 46.0.33445485745541531749546267414718227385437022890768869389066429180265423777079649185104842349209411.1

• Label: 46.0.334...411.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-3.345\times 10^{97}$
• Root discriminant: \(131.85\)
• Ramified primes: $11,47$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - 21*x^45 + 234*x^44 - 1813*x^43 + 11104*x^42 - 57853*x^41 + 268455*x^40 - 1137429*x^39 + 4465637*x^38 - 16402546*x^37 + 56828720*x^36 - 186820582*x^35 + 585518431*x^34 - 1755193762*x^33 + 5047798326*x^32 - 13956961576*x^31 + 37178279250*x^30 - 95529053696*x^29 + 237111330579*x^28 - 568897301282*x^27 + 1320756058230*x^26 - 2967568274611*x^25 + 6457810749477*x^24 - 13607127733128*x^23 + 27776633354903*x^22 - 54891520145448*x^21 + 105060081565914*x^20 - 194491671575647*x^19 + 348417978801750*x^18 - 602707519775443*x^17 + 1007440894372913*x^16 - 1621717831102939*x^15 + 2517231470674524*x^14 - 3747360102999732*x^13 + 5363969908946175*x^12 - 7317027854923049*x^11 + 9562510933179307*x^10 - 11787888134809172*x^9 + 13862890922775304*x^8 - 15106564909634047*x^7 + 15643485715141214*x^6 - 14502253394888869*x^5 + 12777846887095319*x^4 - 9315770272382043*x^3 + 6584077168581835*x^2 - 3020416683060232*x + 1523302345994551

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		-33445485745541531749546267414718227385437022890768869389066429180265423777079649185104842349209411
Root discriminant:		\(131.85\)
Galois root discriminant:		$11^{1/2}47^{22/23}\approx 131.85429884108746$
Ramified primes:		\(11\), \(47\)
Discriminant root field:		\(\Q(\sqrt{-11}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(517=11\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{517}(384,·)$, $\chi_{517}(1,·)$, $\chi_{517}(131,·)$, $\chi_{517}(263,·)$, $\chi_{517}(12,·)$, $\chi_{517}(397,·)$, $\chi_{517}(142,·)$, $\chi_{517}(144,·)$, $\chi_{517}(21,·)$, $\chi_{517}(408,·)$, $\chi_{517}(153,·)$, $\chi_{517}(155,·)$, $\chi_{517}(285,·)$, $\chi_{517}(32,·)$, $\chi_{517}(34,·)$, $\chi_{517}(166,·)$, $\chi_{517}(296,·)$, $\chi_{517}(298,·)$, $\chi_{517}(430,·)$, $\chi_{517}(175,·)$, $\chi_{517}(177,·)$, $\chi_{517}(307,·)$, $\chi_{517}(309,·)$, $\chi_{517}(54,·)$, $\chi_{517}(439,·)$, $\chi_{517}(56,·)$, $\chi_{517}(441,·)$, $\chi_{517}(318,·)$, $\chi_{517}(65,·)$, $\chi_{517}(450,·)$, $\chi_{517}(197,·)$, $\chi_{517}(331,·)$, $\chi_{517}(472,·)$, $\chi_{517}(89,·)$, $\chi_{517}(474,·)$, $\chi_{517}(353,·)$, $\chi_{517}(98,·)$, $\chi_{517}(100,·)$, $\chi_{517}(230,·)$, $\chi_{517}(494,·)$, $\chi_{517}(111,·)$, $\chi_{517}(241,·)$, $\chi_{517}(243,·)$, $\chi_{517}(122,·)$, $\chi_{517}(507,·)$, $\chi_{517}(252,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $\frac{1}{11\!\cdots\!79}a^{45}-\frac{16\!\cdots\!40}{11\!\cdots\!79}a^{44}-\frac{10\!\cdots\!83}{11\!\cdots\!79}a^{43}+\frac{47\!\cdots\!69}{11\!\cdots\!79}a^{42}-\frac{49\!\cdots\!42}{11\!\cdots\!79}a^{41}-\frac{40\!\cdots\!25}{11\!\cdots\!79}a^{40}-\frac{18\!\cdots\!65}{11\!\cdots\!79}a^{39}-\frac{80\!\cdots\!07}{11\!\cdots\!79}a^{38}-\frac{38\!\cdots\!94}{11\!\cdots\!79}a^{37}+\frac{26\!\cdots\!02}{11\!\cdots\!79}a^{36}+\frac{38\!\cdots\!24}{11\!\cdots\!79}a^{35}+\frac{13\!\cdots\!68}{11\!\cdots\!79}a^{34}+\frac{14\!\cdots\!72}{11\!\cdots\!79}a^{33}-\frac{33\!\cdots\!68}{11\!\cdots\!79}a^{32}+\frac{25\!\cdots\!75}{11\!\cdots\!79}a^{31}-\frac{18\!\cdots\!92}{11\!\cdots\!79}a^{30}-\frac{10\!\cdots\!10}{11\!\cdots\!79}a^{29}-\frac{15\!\cdots\!77}{11\!\cdots\!79}a^{28}+\frac{78\!\cdots\!73}{11\!\cdots\!79}a^{27}-\frac{89\!\cdots\!40}{11\!\cdots\!79}a^{26}-\frac{20\!\cdots\!57}{11\!\cdots\!79}a^{25}-\frac{13\!\cdots\!32}{11\!\cdots\!79}a^{24}-\frac{42\!\cdots\!08}{11\!\cdots\!79}a^{23}-\frac{12\!\cdots\!04}{11\!\cdots\!79}a^{22}+\frac{34\!\cdots\!67}{11\!\cdots\!79}a^{21}+\frac{45\!\cdots\!64}{11\!\cdots\!79}a^{20}-\frac{49\!\cdots\!74}{11\!\cdots\!79}a^{19}-\frac{19\!\cdots\!94}{11\!\cdots\!79}a^{18}+\frac{15\!\cdots\!66}{11\!\cdots\!79}a^{17}-\frac{64\!\cdots\!78}{11\!\cdots\!79}a^{16}-\frac{16\!\cdots\!72}{11\!\cdots\!79}a^{15}+\frac{20\!\cdots\!48}{11\!\cdots\!79}a^{14}+\frac{39\!\cdots\!14}{11\!\cdots\!79}a^{13}-\frac{11\!\cdots\!23}{11\!\cdots\!79}a^{12}-\frac{26\!\cdots\!49}{11\!\cdots\!79}a^{11}+\frac{28\!\cdots\!13}{11\!\cdots\!79}a^{10}+\frac{40\!\cdots\!33}{11\!\cdots\!79}a^{9}+\frac{26\!\cdots\!32}{11\!\cdots\!79}a^{8}+\frac{85\!\cdots\!42}{11\!\cdots\!79}a^{7}+\frac{18\!\cdots\!25}{11\!\cdots\!79}a^{6}-\frac{20\!\cdots\!78}{11\!\cdots\!79}a^{5}-\frac{49\!\cdots\!64}{11\!\cdots\!79}a^{4}-\frac{30\!\cdots\!13}{11\!\cdots\!79}a^{3}-\frac{21\!\cdots\!67}{11\!\cdots\!79}a^{2}+\frac{29\!\cdots\!00}{11\!\cdots\!79}a-\frac{31\!\cdots\!69}{11\!\cdots\!79}$

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

Class group and class number

not computed

Unit group

Rank:		$22$
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

$C_{46}$ (as 46T1):

A cyclic group of order 46

The 46 conjugacy class representatives for $C_{46}$

Character table for $C_{46}$

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{47})^+\)

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	$46$	$23^{2}$	$23^{2}$	$46$	R	$46$	$46$	$46$	$23^{2}$	$46$	$23^{2}$	$23^{2}$	$46$	$46$	R	$23^{2}$	$23^{2}$

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
\(11\)		Deg $46$	$2$	$23$	$23$
\(47\)	47.23.22.1	$x^{23} + 47$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$
	47.23.22.1	$x^{23} + 47$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$