Normalized defining polynomial
\( x^{44} - x^{43} - 129 x^{42} + 162 x^{41} + 7330 x^{40} - 10994 x^{39} - 242871 x^{38} + 421787 x^{37} + 5231229 x^{36} - 10324428 x^{35} - 77259269 x^{34} + \cdots - 215069 \)
Invariants
Degree: | $44$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[44, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(576\!\cdots\!125\) \(\medspace = 5^{33}\cdot 67^{42}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(185.05\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $5^{3/4}67^{21/22}\approx 185.05418274979348$ | ||
Ramified primes: | \(5\), \(67\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(335=5\cdot 67\) | ||
Dirichlet character group: | $\lbrace$$\chi_{335}(1,·)$, $\chi_{335}(3,·)$, $\chi_{335}(133,·)$, $\chi_{335}(129,·)$, $\chi_{335}(8,·)$, $\chi_{335}(9,·)$, $\chi_{335}(267,·)$, $\chi_{335}(269,·)$, $\chi_{335}(14,·)$, $\chi_{335}(273,·)$, $\chi_{335}(131,·)$, $\chi_{335}(149,·)$, $\chi_{335}(24,·)$, $\chi_{335}(27,·)$, $\chi_{335}(156,·)$, $\chi_{335}(159,·)$, $\chi_{335}(64,·)$, $\chi_{335}(42,·)$, $\chi_{335}(43,·)$, $\chi_{335}(174,·)$, $\chi_{335}(177,·)$, $\chi_{335}(52,·)$, $\chi_{335}(53,·)$, $\chi_{335}(137,·)$, $\chi_{335}(313,·)$, $\chi_{335}(58,·)$, $\chi_{335}(59,·)$, $\chi_{335}(192,·)$, $\chi_{335}(196,·)$, $\chi_{335}(72,·)$, $\chi_{335}(76,·)$, $\chi_{335}(81,·)$, $\chi_{335}(142,·)$, $\chi_{335}(216,·)$, $\chi_{335}(89,·)$, $\chi_{335}(91,·)$, $\chi_{335}(226,·)$, $\chi_{335}(187,·)$, $\chi_{335}(228,·)$, $\chi_{335}(112,·)$, $\chi_{335}(241,·)$, $\chi_{335}(243,·)$, $\chi_{335}(253,·)$, $\chi_{335}(126,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{75\!\cdots\!91}a^{42}-\frac{30\!\cdots\!27}{75\!\cdots\!91}a^{41}-\frac{35\!\cdots\!02}{75\!\cdots\!91}a^{40}-\frac{12\!\cdots\!61}{75\!\cdots\!91}a^{39}-\frac{33\!\cdots\!38}{75\!\cdots\!91}a^{38}+\frac{32\!\cdots\!49}{75\!\cdots\!91}a^{37}-\frac{10\!\cdots\!02}{75\!\cdots\!91}a^{36}-\frac{12\!\cdots\!35}{75\!\cdots\!91}a^{35}+\frac{41\!\cdots\!77}{75\!\cdots\!91}a^{34}-\frac{90\!\cdots\!56}{75\!\cdots\!91}a^{33}-\frac{52\!\cdots\!80}{75\!\cdots\!91}a^{32}-\frac{25\!\cdots\!89}{75\!\cdots\!91}a^{31}+\frac{28\!\cdots\!74}{75\!\cdots\!91}a^{30}-\frac{30\!\cdots\!35}{75\!\cdots\!91}a^{29}+\frac{50\!\cdots\!40}{75\!\cdots\!91}a^{28}+\frac{10\!\cdots\!27}{75\!\cdots\!91}a^{27}+\frac{37\!\cdots\!59}{75\!\cdots\!91}a^{26}-\frac{27\!\cdots\!05}{75\!\cdots\!91}a^{25}-\frac{15\!\cdots\!12}{75\!\cdots\!91}a^{24}-\frac{32\!\cdots\!41}{75\!\cdots\!91}a^{23}-\frac{32\!\cdots\!68}{75\!\cdots\!91}a^{22}-\frac{35\!\cdots\!06}{75\!\cdots\!91}a^{21}+\frac{11\!\cdots\!09}{75\!\cdots\!91}a^{20}+\frac{26\!\cdots\!41}{75\!\cdots\!91}a^{19}+\frac{28\!\cdots\!42}{75\!\cdots\!91}a^{18}-\frac{35\!\cdots\!22}{75\!\cdots\!91}a^{17}-\frac{21\!\cdots\!88}{75\!\cdots\!91}a^{16}-\frac{33\!\cdots\!47}{75\!\cdots\!91}a^{15}+\frac{16\!\cdots\!03}{75\!\cdots\!91}a^{14}-\frac{14\!\cdots\!78}{75\!\cdots\!91}a^{13}+\frac{10\!\cdots\!14}{75\!\cdots\!91}a^{12}+\frac{16\!\cdots\!14}{75\!\cdots\!91}a^{11}+\frac{28\!\cdots\!67}{75\!\cdots\!91}a^{10}-\frac{13\!\cdots\!14}{75\!\cdots\!91}a^{9}-\frac{11\!\cdots\!90}{75\!\cdots\!91}a^{8}+\frac{15\!\cdots\!20}{75\!\cdots\!91}a^{7}+\frac{94\!\cdots\!10}{75\!\cdots\!91}a^{6}+\frac{21\!\cdots\!58}{75\!\cdots\!91}a^{5}+\frac{37\!\cdots\!95}{75\!\cdots\!91}a^{4}+\frac{31\!\cdots\!10}{75\!\cdots\!91}a^{3}+\frac{68\!\cdots\!76}{75\!\cdots\!91}a^{2}-\frac{42\!\cdots\!76}{75\!\cdots\!91}a-\frac{25\!\cdots\!39}{75\!\cdots\!91}$, $\frac{1}{10\!\cdots\!51}a^{43}-\frac{23\!\cdots\!40}{10\!\cdots\!51}a^{42}-\frac{32\!\cdots\!74}{10\!\cdots\!51}a^{41}+\frac{23\!\cdots\!65}{10\!\cdots\!51}a^{40}+\frac{28\!\cdots\!92}{10\!\cdots\!51}a^{39}+\frac{20\!\cdots\!92}{10\!\cdots\!51}a^{38}+\frac{43\!\cdots\!42}{10\!\cdots\!51}a^{37}+\frac{32\!\cdots\!15}{10\!\cdots\!51}a^{36}-\frac{23\!\cdots\!81}{10\!\cdots\!51}a^{35}-\frac{37\!\cdots\!06}{10\!\cdots\!51}a^{34}+\frac{52\!\cdots\!59}{10\!\cdots\!51}a^{33}+\frac{51\!\cdots\!07}{10\!\cdots\!51}a^{32}+\frac{37\!\cdots\!72}{10\!\cdots\!51}a^{31}-\frac{10\!\cdots\!66}{10\!\cdots\!51}a^{30}+\frac{28\!\cdots\!23}{10\!\cdots\!51}a^{29}+\frac{36\!\cdots\!85}{10\!\cdots\!51}a^{28}+\frac{39\!\cdots\!43}{10\!\cdots\!51}a^{27}-\frac{41\!\cdots\!37}{10\!\cdots\!51}a^{26}+\frac{18\!\cdots\!79}{10\!\cdots\!51}a^{25}-\frac{51\!\cdots\!25}{10\!\cdots\!51}a^{24}-\frac{13\!\cdots\!93}{10\!\cdots\!51}a^{23}-\frac{18\!\cdots\!81}{10\!\cdots\!51}a^{22}-\frac{34\!\cdots\!40}{10\!\cdots\!51}a^{21}-\frac{28\!\cdots\!72}{10\!\cdots\!51}a^{20}-\frac{21\!\cdots\!64}{10\!\cdots\!51}a^{19}+\frac{11\!\cdots\!69}{10\!\cdots\!51}a^{18}+\frac{26\!\cdots\!25}{10\!\cdots\!51}a^{17}+\frac{63\!\cdots\!18}{10\!\cdots\!51}a^{16}-\frac{35\!\cdots\!45}{10\!\cdots\!51}a^{15}-\frac{18\!\cdots\!12}{10\!\cdots\!51}a^{14}-\frac{32\!\cdots\!50}{10\!\cdots\!51}a^{13}+\frac{34\!\cdots\!53}{10\!\cdots\!51}a^{12}+\frac{63\!\cdots\!16}{10\!\cdots\!51}a^{11}-\frac{61\!\cdots\!48}{10\!\cdots\!51}a^{10}+\frac{30\!\cdots\!33}{10\!\cdots\!51}a^{9}+\frac{12\!\cdots\!87}{10\!\cdots\!51}a^{8}+\frac{16\!\cdots\!61}{10\!\cdots\!51}a^{7}-\frac{31\!\cdots\!31}{10\!\cdots\!51}a^{6}-\frac{14\!\cdots\!77}{10\!\cdots\!51}a^{5}+\frac{50\!\cdots\!81}{10\!\cdots\!51}a^{4}-\frac{34\!\cdots\!07}{10\!\cdots\!51}a^{3}-\frac{98\!\cdots\!81}{10\!\cdots\!51}a^{2}+\frac{50\!\cdots\!48}{10\!\cdots\!51}a-\frac{70\!\cdots\!61}{21\!\cdots\!49}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $43$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
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 $
Galois group
A cyclic group of order 44 |
The 44 conjugacy class representatives for $C_{44}$ |
Character table for $C_{44}$ is not computed |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.4.561125.1, 11.11.1822837804551761449.1, 22.22.162243049887845980095628744560672832080078125.1 |
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 | $44$ | $44$ | R | $44$ | $22^{2}$ | $44$ | $44$ | $22^{2}$ | $44$ | ${\href{/padicField/29.2.0.1}{2} }^{22}$ | $22^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{11}$ | $22^{2}$ | $44$ | $44$ | $44$ | $22^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | Deg $44$ | $4$ | $11$ | $33$ | |||
\(67\) | Deg $44$ | $22$ | $2$ | $42$ |