Normalized defining polynomial
\( x^{45} - 87 x^{43} - 4 x^{42} + 3402 x^{41} + 300 x^{40} - 79106 x^{39} - 10008 x^{38} + 1219668 x^{37} + \cdots + 1 \)
Invariants
| Degree: | $45$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[45, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(183\!\cdots\!361\)
\(\medspace = 3^{60}\cdot 31^{42}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(106.68\) | 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: | $3^{4/3}31^{14/15}\approx 106.68399094818109$ | ||
| Ramified primes: |
\(3\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $45$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(279=3^{2}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{279}(256,·)$, $\chi_{279}(1,·)$, $\chi_{279}(4,·)$, $\chi_{279}(133,·)$, $\chi_{279}(262,·)$, $\chi_{279}(7,·)$, $\chi_{279}(10,·)$, $\chi_{279}(268,·)$, $\chi_{279}(142,·)$, $\chi_{279}(16,·)$, $\chi_{279}(19,·)$, $\chi_{279}(25,·)$, $\chi_{279}(28,·)$, $\chi_{279}(157,·)$, $\chi_{279}(160,·)$, $\chi_{279}(163,·)$, $\chi_{279}(40,·)$, $\chi_{279}(169,·)$, $\chi_{279}(175,·)$, $\chi_{279}(49,·)$, $\chi_{279}(187,·)$, $\chi_{279}(190,·)$, $\chi_{279}(64,·)$, $\chi_{279}(193,·)$, $\chi_{279}(67,·)$, $\chi_{279}(196,·)$, $\chi_{279}(70,·)$, $\chi_{279}(202,·)$, $\chi_{279}(76,·)$, $\chi_{279}(205,·)$, $\chi_{279}(82,·)$, $\chi_{279}(211,·)$, $\chi_{279}(214,·)$, $\chi_{279}(94,·)$, $\chi_{279}(97,·)$, $\chi_{279}(226,·)$, $\chi_{279}(100,·)$, $\chi_{279}(103,·)$, $\chi_{279}(235,·)$, $\chi_{279}(109,·)$, $\chi_{279}(112,·)$, $\chi_{279}(118,·)$, $\chi_{279}(121,·)$, $\chi_{279}(250,·)$, $\chi_{279}(253,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{5}a^{42}-\frac{1}{5}a^{41}+\frac{1}{5}a^{40}-\frac{1}{5}a^{39}-\frac{1}{5}a^{38}+\frac{1}{5}a^{36}-\frac{2}{5}a^{32}-\frac{2}{5}a^{29}+\frac{2}{5}a^{27}+\frac{2}{5}a^{24}+\frac{1}{5}a^{20}-\frac{1}{5}a^{19}+\frac{1}{5}a^{18}+\frac{2}{5}a^{17}-\frac{2}{5}a^{15}+\frac{1}{5}a^{14}-\frac{1}{5}a^{12}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{2785}a^{43}+\frac{35}{557}a^{42}+\frac{161}{557}a^{41}-\frac{189}{557}a^{40}+\frac{453}{2785}a^{39}-\frac{256}{2785}a^{38}-\frac{484}{2785}a^{37}-\frac{889}{2785}a^{36}+\frac{180}{557}a^{35}+\frac{277}{557}a^{34}+\frac{598}{2785}a^{33}+\frac{1363}{2785}a^{32}-\frac{160}{557}a^{31}-\frac{1182}{2785}a^{30}-\frac{892}{2785}a^{29}-\frac{928}{2785}a^{28}+\frac{392}{2785}a^{27}-\frac{23}{557}a^{26}+\frac{1367}{2785}a^{25}+\frac{827}{2785}a^{24}-\frac{106}{557}a^{23}+\frac{175}{557}a^{22}+\frac{606}{2785}a^{21}+\frac{75}{557}a^{20}+\frac{167}{557}a^{19}+\frac{888}{2785}a^{18}-\frac{938}{2785}a^{17}+\frac{8}{2785}a^{16}-\frac{206}{2785}a^{15}-\frac{764}{2785}a^{14}-\frac{296}{2785}a^{13}-\frac{31}{2785}a^{12}+\frac{85}{557}a^{11}-\frac{1289}{2785}a^{10}+\frac{1302}{2785}a^{9}-\frac{726}{2785}a^{8}+\frac{704}{2785}a^{7}-\frac{1012}{2785}a^{6}-\frac{1162}{2785}a^{5}+\frac{1013}{2785}a^{4}-\frac{403}{2785}a^{3}+\frac{508}{2785}a^{2}+\frac{137}{2785}a+\frac{134}{2785}$, $\frac{1}{66\!\cdots\!65}a^{44}-\frac{52\!\cdots\!62}{66\!\cdots\!65}a^{43}-\frac{55\!\cdots\!32}{66\!\cdots\!65}a^{42}-\frac{10\!\cdots\!78}{66\!\cdots\!65}a^{41}+\frac{61\!\cdots\!06}{66\!\cdots\!65}a^{40}+\frac{46\!\cdots\!61}{13\!\cdots\!73}a^{39}-\frac{33\!\cdots\!12}{13\!\cdots\!73}a^{38}+\frac{29\!\cdots\!74}{66\!\cdots\!65}a^{37}+\frac{66\!\cdots\!16}{66\!\cdots\!65}a^{36}-\frac{50\!\cdots\!90}{13\!\cdots\!73}a^{35}+\frac{27\!\cdots\!88}{66\!\cdots\!65}a^{34}+\frac{80\!\cdots\!22}{66\!\cdots\!65}a^{33}+\frac{30\!\cdots\!58}{66\!\cdots\!65}a^{32}+\frac{29\!\cdots\!98}{66\!\cdots\!65}a^{31}+\frac{87\!\cdots\!32}{66\!\cdots\!65}a^{30}+\frac{36\!\cdots\!52}{13\!\cdots\!73}a^{29}-\frac{28\!\cdots\!02}{66\!\cdots\!65}a^{28}-\frac{34\!\cdots\!53}{66\!\cdots\!65}a^{27}-\frac{16\!\cdots\!17}{43\!\cdots\!85}a^{26}-\frac{32\!\cdots\!77}{66\!\cdots\!65}a^{25}+\frac{10\!\cdots\!72}{66\!\cdots\!65}a^{24}+\frac{34\!\cdots\!85}{13\!\cdots\!73}a^{23}+\frac{26\!\cdots\!66}{66\!\cdots\!65}a^{22}+\frac{26\!\cdots\!93}{66\!\cdots\!65}a^{21}+\frac{32\!\cdots\!03}{66\!\cdots\!65}a^{20}-\frac{20\!\cdots\!45}{13\!\cdots\!73}a^{19}+\frac{30\!\cdots\!04}{66\!\cdots\!65}a^{18}+\frac{29\!\cdots\!07}{13\!\cdots\!73}a^{17}-\frac{10\!\cdots\!82}{66\!\cdots\!65}a^{16}+\frac{25\!\cdots\!17}{66\!\cdots\!65}a^{15}-\frac{42\!\cdots\!19}{13\!\cdots\!73}a^{14}-\frac{60\!\cdots\!04}{66\!\cdots\!65}a^{13}+\frac{13\!\cdots\!79}{66\!\cdots\!65}a^{12}-\frac{12\!\cdots\!54}{66\!\cdots\!65}a^{11}-\frac{44\!\cdots\!75}{13\!\cdots\!73}a^{10}+\frac{12\!\cdots\!88}{66\!\cdots\!65}a^{9}-\frac{69\!\cdots\!91}{66\!\cdots\!65}a^{8}+\frac{23\!\cdots\!44}{66\!\cdots\!65}a^{7}-\frac{21\!\cdots\!01}{13\!\cdots\!73}a^{6}+\frac{20\!\cdots\!03}{66\!\cdots\!65}a^{5}+\frac{12\!\cdots\!54}{66\!\cdots\!65}a^{4}-\frac{45\!\cdots\!24}{13\!\cdots\!73}a^{3}+\frac{10\!\cdots\!21}{66\!\cdots\!65}a^{2}+\frac{14\!\cdots\!99}{66\!\cdots\!65}a+\frac{11\!\cdots\!74}{66\!\cdots\!65}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $44$ | 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
$C_3\times C_{15}$ (as 45T2):
| An abelian group of order 45 |
| The 45 conjugacy class representatives for $C_3\times C_{15}$ |
| Character table for $C_3\times C_{15}$ |
Intermediate fields
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 | $15^{3}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{15}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | R | ${\href{/padicField/37.3.0.1}{3} }^{15}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $45$ | $3$ | $15$ | $60$ | |||
|
\(31\)
| Deg $45$ | $15$ | $3$ | $42$ |