Global number field 46.46.1472629538247713057115908841446795301275975338441612336549649339098854081833834905600000000000000000000000.1

• Label: 46.46.1472629538...0000.1
• Degree: $46$
• Signature: $[46, 0]$
• Discriminant: $2^{46}\cdot 5^{23}\cdot 47^{45}$
• Root discriminant: $193.31$
• Ramified primes: $2, 5, 47$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

\( x^{46} - 235 x^{44} + 25850 x^{42} - 1768375 x^{40} + 84306250 x^{38} - 2974806250 x^{36} + 80537437500 x^{34} - 1711420546875 x^{32} + 28962501562500 x^{30} - 393788398437500 x^{28} + 4321029453125000 x^{26} - 38300033789062500 x^{24} + 273571669921875000 x^{22} - 1565919287109375000 x^{20} + 7117814941406250000 x^{18} - 25357215728759765625 x^{16} + 69527849578857421875 x^{14} - 143145572662353515625 x^{12} + 213895683288574218750 x^{10} - 221132755279541015625 x^{8} + 147421836853027343750 x^{6} - 56700706481933593750 x^{4} + 10309219360351562500 x^{2} - 560283660888671875 \)

Invariants

Degree:		$46$
Signature:		$[46, 0]$
Discriminant:		\(1472629538247713057115908841446795301275975338441612336549649339098854081833834905600000000000000000000000=2^{46}\cdot 5^{23}\cdot 47^{45}\)
Root discriminant:		$193.31$
Ramified primes:		$2, 5, 47$
This field is Galois and abelian over $\Q$.
Conductor:		\(940=2^{2}\cdot 5\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{940}(1,·)$, $\chi_{940}(901,·)$, $\chi_{940}(521,·)$, $\chi_{940}(139,·)$, $\chi_{940}(399,·)$, $\chi_{940}(401,·)$, $\chi_{940}(19,·)$, $\chi_{940}(21,·)$, $\chi_{940}(279,·)$, $\chi_{940}(921,·)$, $\chi_{940}(539,·)$, $\chi_{940}(541,·)$, $\chi_{940}(801,·)$, $\chi_{940}(419,·)$, $\chi_{940}(39,·)$, $\chi_{940}(939,·)$, $\chi_{940}(179,·)$, $\chi_{940}(121,·)$, $\chi_{940}(441,·)$, $\chi_{940}(699,·)$, $\chi_{940}(61,·)$, $\chi_{940}(579,·)$, $\chi_{940}(581,·)$, $\chi_{940}(839,·)$, $\chi_{940}(841,·)$, $\chi_{940}(859,·)$, $\chi_{940}(721,·)$, $\chi_{940}(339,·)$, $\chi_{940}(919,·)$, $\chi_{940}(341,·)$, $\chi_{940}(599,·)$, $\chi_{940}(601,·)$, $\chi_{940}(219,·)$, $\chi_{940}(199,·)$, $\chi_{940}(101,·)$, $\chi_{940}(99,·)$, $\chi_{940}(741,·)$, $\chi_{940}(81,·)$, $\chi_{940}(361,·)$, $\chi_{940}(359,·)$, $\chi_{940}(879,·)$, $\chi_{940}(241,·)$, $\chi_{940}(499,·)$, $\chi_{940}(761,·)$, $\chi_{940}(819,·)$, $\chi_{940}(661,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{3125} a^{10}$, $\frac{1}{3125} a^{11}$, $\frac{1}{15625} a^{12}$, $\frac{1}{15625} a^{13}$, $\frac{1}{78125} a^{14}$, $\frac{1}{78125} a^{15}$, $\frac{1}{390625} a^{16}$, $\frac{1}{390625} a^{17}$, $\frac{1}{1953125} a^{18}$, $\frac{1}{1953125} a^{19}$, $\frac{1}{9765625} a^{20}$, $\frac{1}{9765625} a^{21}$, $\frac{1}{48828125} a^{22}$, $\frac{1}{48828125} a^{23}$, $\frac{1}{244140625} a^{24}$, $\frac{1}{244140625} a^{25}$, $\frac{1}{1220703125} a^{26}$, $\frac{1}{1220703125} a^{27}$, $\frac{1}{6103515625} a^{28}$, $\frac{1}{6103515625} a^{29}$, $\frac{1}{30517578125} a^{30}$, $\frac{1}{30517578125} a^{31}$, $\frac{1}{152587890625} a^{32}$, $\frac{1}{152587890625} a^{33}$, $\frac{1}{762939453125} a^{34}$, $\frac{1}{762939453125} a^{35}$, $\frac{1}{3814697265625} a^{36}$, $\frac{1}{3814697265625} a^{37}$, $\frac{1}{19073486328125} a^{38}$, $\frac{1}{19073486328125} a^{39}$, $\frac{1}{95367431640625} a^{40}$, $\frac{1}{95367431640625} a^{41}$, $\frac{1}{476837158203125} a^{42}$, $\frac{1}{476837158203125} a^{43}$, $\frac{1}{2384185791015625} a^{44}$, $\frac{1}{2384185791015625} a^{45}$

Class group and class number

Not computed

Unit group

Rank:		$45$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Not computed
Regulator:		Not computed

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

\(\Q(\sqrt{235}) \), \(\Q(\zeta_{47})^+\)

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	R	$23^{2}$	R	$23^{2}$	$23^{2}$	$23^{2}$	$46$	$23^{2}$	$46$	$46$	$23^{2}$	$46$	$46$	$46$	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
2	Data not computed
5	Data not computed
47	Data not computed