Number field 46.0.9630272387131177947845223207051634018451530879887040478778044189300817518386752324750061473265804243579.1

• Label: 46.0.963...579.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-9.630\times 10^{102}$
• Root discriminant: \(173.29\)
• Ramified primes: $19,47$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		-9630272387131177947845223207051634018451530879887040478778044189300817518386752324750061473265804243579
Root discriminant:		\(173.29\)
Galois root discriminant:		$19^{1/2}47^{22/23}\approx 173.29049869947002$
Ramified primes:		\(19\), \(47\)
Discriminant root field:		\(\Q(\sqrt{-19}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(893=19\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{893}(512,·)$, $\chi_{893}(1,·)$, $\chi_{893}(645,·)$, $\chi_{893}(647,·)$, $\chi_{893}(267,·)$, $\chi_{893}(780,·)$, $\chi_{893}(400,·)$, $\chi_{893}(18,·)$, $\chi_{893}(531,·)$, $\chi_{893}(533,·)$, $\chi_{893}(664,·)$, $\chi_{893}(153,·)$, $\chi_{893}(666,·)$, $\chi_{893}(284,·)$, $\chi_{893}(286,·)$, $\chi_{893}(37,·)$, $\chi_{893}(683,·)$, $\chi_{893}(685,·)$, $\chi_{893}(303,·)$, $\chi_{893}(816,·)$, $\chi_{893}(56,·)$, $\chi_{893}(571,·)$, $\chi_{893}(189,·)$, $\chi_{893}(191,·)$, $\chi_{893}(835,·)$, $\chi_{893}(324,·)$, $\chi_{893}(455,·)$, $\chi_{893}(457,·)$, $\chi_{893}(75,·)$, $\chi_{893}(588,·)$, $\chi_{893}(721,·)$, $\chi_{893}(723,·)$, $\chi_{893}(341,·)$, $\chi_{893}(854,·)$, $\chi_{893}(343,·)$, $\chi_{893}(474,·)$, $\chi_{893}(476,·)$, $\chi_{893}(96,·)$, $\chi_{893}(742,·)$, $\chi_{893}(873,·)$, $\chi_{893}(495,·)$, $\chi_{893}(115,·)$, $\chi_{893}(628,·)$, $\chi_{893}(759,·)$, $\chi_{893}(761,·)$, $\chi_{893}(379,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{292339}a^{44}+\frac{119321}{292339}a^{43}+\frac{11852}{292339}a^{42}+\frac{68871}{292339}a^{41}-\frac{143641}{292339}a^{40}+\frac{68966}{292339}a^{39}-\frac{87443}{292339}a^{38}+\frac{101125}{292339}a^{37}-\frac{4257}{292339}a^{36}-\frac{139659}{292339}a^{35}+\frac{5458}{292339}a^{34}+\frac{17767}{292339}a^{33}+\frac{89815}{292339}a^{32}+\frac{6949}{292339}a^{31}-\frac{82433}{292339}a^{30}+\frac{86838}{292339}a^{29}-\frac{89222}{292339}a^{28}+\frac{141818}{292339}a^{27}-\frac{83610}{292339}a^{26}-\frac{94682}{292339}a^{25}-\frac{58332}{292339}a^{24}-\frac{57862}{292339}a^{23}+\frac{101356}{292339}a^{22}-\frac{97149}{292339}a^{21}+\frac{85478}{292339}a^{20}+\frac{96989}{292339}a^{19}-\frac{94515}{292339}a^{18}+\frac{82398}{292339}a^{17}+\frac{114440}{292339}a^{16}+\frac{136379}{292339}a^{15}-\frac{32246}{292339}a^{14}-\frac{103592}{292339}a^{13}-\frac{81067}{292339}a^{12}+\frac{6318}{292339}a^{11}-\frac{22347}{292339}a^{10}+\frac{143073}{292339}a^{9}-\frac{85888}{292339}a^{8}+\frac{51027}{292339}a^{7}-\frac{129068}{292339}a^{6}+\frac{122798}{292339}a^{5}+\frac{88836}{292339}a^{4}-\frac{60384}{292339}a^{3}-\frac{98040}{292339}a^{2}+\frac{26881}{292339}a+\frac{80}{1033}$, $\frac{1}{44\!\cdots\!69}a^{45}+\frac{52\!\cdots\!20}{44\!\cdots\!69}a^{44}+\frac{16\!\cdots\!44}{44\!\cdots\!69}a^{43}-\frac{16\!\cdots\!06}{44\!\cdots\!69}a^{42}-\frac{20\!\cdots\!27}{44\!\cdots\!69}a^{41}-\frac{18\!\cdots\!09}{44\!\cdots\!69}a^{40}-\frac{75\!\cdots\!48}{44\!\cdots\!69}a^{39}-\frac{19\!\cdots\!56}{44\!\cdots\!69}a^{38}-\frac{98\!\cdots\!77}{44\!\cdots\!69}a^{37}-\frac{58\!\cdots\!57}{44\!\cdots\!69}a^{36}-\frac{92\!\cdots\!77}{44\!\cdots\!69}a^{35}+\frac{11\!\cdots\!73}{44\!\cdots\!69}a^{34}+\frac{19\!\cdots\!51}{44\!\cdots\!69}a^{33}+\frac{13\!\cdots\!69}{44\!\cdots\!69}a^{32}+\frac{16\!\cdots\!26}{44\!\cdots\!69}a^{31}+\frac{25\!\cdots\!36}{44\!\cdots\!69}a^{30}+\frac{18\!\cdots\!34}{44\!\cdots\!69}a^{29}+\frac{11\!\cdots\!59}{44\!\cdots\!69}a^{28}-\frac{43\!\cdots\!12}{44\!\cdots\!69}a^{27}+\frac{18\!\cdots\!72}{44\!\cdots\!69}a^{26}-\frac{13\!\cdots\!17}{44\!\cdots\!69}a^{25}-\frac{22\!\cdots\!50}{44\!\cdots\!69}a^{24}+\frac{80\!\cdots\!47}{44\!\cdots\!69}a^{23}-\frac{20\!\cdots\!39}{44\!\cdots\!69}a^{22}-\frac{83\!\cdots\!10}{44\!\cdots\!69}a^{21}+\frac{26\!\cdots\!37}{44\!\cdots\!69}a^{20}-\frac{57\!\cdots\!41}{44\!\cdots\!69}a^{19}+\frac{20\!\cdots\!99}{44\!\cdots\!69}a^{18}-\frac{45\!\cdots\!01}{44\!\cdots\!69}a^{17}-\frac{14\!\cdots\!40}{44\!\cdots\!69}a^{16}-\frac{41\!\cdots\!22}{44\!\cdots\!69}a^{15}+\frac{20\!\cdots\!38}{44\!\cdots\!69}a^{14}-\frac{74\!\cdots\!91}{44\!\cdots\!69}a^{13}-\frac{14\!\cdots\!16}{44\!\cdots\!69}a^{12}-\frac{98\!\cdots\!30}{44\!\cdots\!69}a^{11}-\frac{69\!\cdots\!64}{44\!\cdots\!69}a^{10}-\frac{60\!\cdots\!58}{44\!\cdots\!69}a^{9}-\frac{56\!\cdots\!11}{44\!\cdots\!69}a^{8}-\frac{19\!\cdots\!40}{44\!\cdots\!69}a^{7}+\frac{10\!\cdots\!63}{44\!\cdots\!69}a^{6}-\frac{18\!\cdots\!64}{44\!\cdots\!69}a^{5}-\frac{18\!\cdots\!35}{44\!\cdots\!69}a^{4}+\frac{14\!\cdots\!36}{44\!\cdots\!69}a^{3}-\frac{12\!\cdots\!93}{44\!\cdots\!69}a^{2}+\frac{10\!\cdots\!29}{44\!\cdots\!69}a+\frac{48\!\cdots\!34}{15\!\cdots\!43}$

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 $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\zeta_{47})^+\)

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

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
\(19\)		Deg $46$	$2$	$23$	$23$
\(47\)	47.23.22.1	$x^{23} + 47$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$
	47.23.22.1	$x^{23} + 47$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$