Normalized defining polynomial
\( x^{45} - x^{44} - 250 x^{43} - 57 x^{42} + 27766 x^{41} + 34148 x^{40} - 1801054 x^{39} + \cdots - 41375255409223 \)
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: |
\(612\!\cdots\!361\)
\(\medspace = 19^{40}\cdot 31^{42}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(337.76\) | 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: | $19^{8/9}31^{14/15}\approx 337.75940307002355$ | ||
| Ramified primes: |
\(19\), \(31\)
| 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: | \(589=19\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(386,·)$, $\chi_{589}(5,·)$, $\chi_{589}(64,·)$, $\chi_{589}(140,·)$, $\chi_{589}(25,·)$, $\chi_{589}(537,·)$, $\chi_{589}(28,·)$, $\chi_{589}(541,·)$, $\chi_{589}(159,·)$, $\chi_{589}(163,·)$, $\chi_{589}(36,·)$, $\chi_{589}(422,·)$, $\chi_{589}(39,·)$, $\chi_{589}(555,·)$, $\chi_{589}(562,·)$, $\chi_{589}(180,·)$, $\chi_{589}(438,·)$, $\chi_{589}(567,·)$, $\chi_{589}(568,·)$, $\chi_{589}(441,·)$, $\chi_{589}(543,·)$, $\chi_{589}(320,·)$, $\chi_{589}(195,·)$, $\chi_{589}(454,·)$, $\chi_{589}(328,·)$, $\chi_{589}(329,·)$, $\chi_{589}(311,·)$, $\chi_{589}(206,·)$, $\chi_{589}(419,·)$, $\chi_{589}(462,·)$, $\chi_{589}(343,·)$, $\chi_{589}(348,·)$, $\chi_{589}(349,·)$, $\chi_{589}(479,·)$, $\chi_{589}(226,·)$, $\chi_{589}(484,·)$, $\chi_{589}(359,·)$, $\chi_{589}(423,·)$, $\chi_{589}(111,·)$, $\chi_{589}(467,·)$, $\chi_{589}(118,·)$, $\chi_{589}(503,·)$, $\chi_{589}(377,·)$, $\chi_{589}(125,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{683}a^{42}+\frac{51}{683}a^{41}+\frac{237}{683}a^{40}+\frac{84}{683}a^{39}-\frac{112}{683}a^{38}-\frac{298}{683}a^{37}-\frac{273}{683}a^{36}-\frac{147}{683}a^{35}+\frac{238}{683}a^{34}+\frac{5}{683}a^{33}-\frac{81}{683}a^{32}+\frac{23}{683}a^{31}+\frac{215}{683}a^{30}+\frac{59}{683}a^{29}+\frac{158}{683}a^{28}-\frac{280}{683}a^{27}+\frac{15}{683}a^{26}+\frac{262}{683}a^{25}+\frac{323}{683}a^{24}+\frac{116}{683}a^{23}+\frac{9}{683}a^{22}+\frac{262}{683}a^{21}-\frac{37}{683}a^{20}-\frac{229}{683}a^{19}-\frac{140}{683}a^{18}+\frac{40}{683}a^{17}-\frac{96}{683}a^{16}+\frac{99}{683}a^{15}+\frac{98}{683}a^{14}+\frac{202}{683}a^{13}+\frac{224}{683}a^{12}-\frac{163}{683}a^{11}+\frac{192}{683}a^{10}+\frac{131}{683}a^{9}+\frac{134}{683}a^{8}-\frac{204}{683}a^{7}-\frac{56}{683}a^{6}+\frac{146}{683}a^{5}+\frac{274}{683}a^{4}-\frac{263}{683}a^{3}+\frac{307}{683}a^{2}+\frac{157}{683}a+\frac{318}{683}$, $\frac{1}{1574364859}a^{43}+\frac{333299}{1574364859}a^{42}+\frac{728769309}{1574364859}a^{41}-\frac{591194083}{1574364859}a^{40}-\frac{645346928}{1574364859}a^{39}+\frac{137860645}{1574364859}a^{38}-\frac{636292339}{1574364859}a^{37}-\frac{412591989}{1574364859}a^{36}-\frac{724952318}{1574364859}a^{35}+\frac{198926136}{1574364859}a^{34}-\frac{543752370}{1574364859}a^{33}+\frac{41449682}{1574364859}a^{32}+\frac{109190820}{1574364859}a^{31}+\frac{693938558}{1574364859}a^{30}-\frac{402673992}{1574364859}a^{29}-\frac{116285780}{1574364859}a^{28}+\frac{680468788}{1574364859}a^{27}-\frac{547552116}{1574364859}a^{26}-\frac{391147276}{1574364859}a^{25}-\frac{776346507}{1574364859}a^{24}-\frac{631773291}{1574364859}a^{23}-\frac{51178115}{1574364859}a^{22}-\frac{80340973}{1574364859}a^{21}-\frac{539634408}{1574364859}a^{20}+\frac{203026238}{1574364859}a^{19}-\frac{549169198}{1574364859}a^{18}-\frac{495481271}{1574364859}a^{17}-\frac{485285832}{1574364859}a^{16}+\frac{777681576}{1574364859}a^{15}+\frac{722271312}{1574364859}a^{14}-\frac{54359447}{1574364859}a^{13}-\frac{447374975}{1574364859}a^{12}+\frac{681982088}{1574364859}a^{11}-\frac{441100217}{1574364859}a^{10}-\frac{132506470}{1574364859}a^{9}+\frac{512390503}{1574364859}a^{8}-\frac{211235068}{1574364859}a^{7}-\frac{420491815}{1574364859}a^{6}-\frac{328763122}{1574364859}a^{5}-\frac{156655}{589429}a^{4}+\frac{515601490}{1574364859}a^{3}-\frac{472539657}{1574364859}a^{2}-\frac{483856602}{1574364859}a-\frac{314100139}{1574364859}$, $\frac{1}{42\!\cdots\!21}a^{44}-\frac{44\!\cdots\!17}{42\!\cdots\!21}a^{43}+\frac{29\!\cdots\!41}{42\!\cdots\!21}a^{42}+\frac{14\!\cdots\!68}{42\!\cdots\!21}a^{41}-\frac{19\!\cdots\!16}{42\!\cdots\!21}a^{40}-\frac{14\!\cdots\!99}{42\!\cdots\!21}a^{39}+\frac{20\!\cdots\!84}{42\!\cdots\!21}a^{38}+\frac{42\!\cdots\!74}{42\!\cdots\!21}a^{37}+\frac{12\!\cdots\!99}{42\!\cdots\!21}a^{36}-\frac{12\!\cdots\!54}{42\!\cdots\!21}a^{35}-\frac{19\!\cdots\!04}{42\!\cdots\!21}a^{34}-\frac{33\!\cdots\!57}{42\!\cdots\!21}a^{33}-\frac{10\!\cdots\!93}{42\!\cdots\!21}a^{32}-\frac{20\!\cdots\!47}{42\!\cdots\!21}a^{31}+\frac{31\!\cdots\!61}{42\!\cdots\!21}a^{30}-\frac{21\!\cdots\!31}{42\!\cdots\!21}a^{29}-\frac{65\!\cdots\!12}{42\!\cdots\!21}a^{28}+\frac{52\!\cdots\!62}{42\!\cdots\!21}a^{27}-\frac{11\!\cdots\!08}{42\!\cdots\!21}a^{26}+\frac{21\!\cdots\!17}{42\!\cdots\!21}a^{25}+\frac{68\!\cdots\!47}{42\!\cdots\!21}a^{24}-\frac{20\!\cdots\!83}{42\!\cdots\!21}a^{23}+\frac{13\!\cdots\!84}{42\!\cdots\!21}a^{22}+\frac{18\!\cdots\!96}{42\!\cdots\!21}a^{21}+\frac{48\!\cdots\!53}{42\!\cdots\!21}a^{20}-\frac{15\!\cdots\!63}{42\!\cdots\!21}a^{19}-\frac{53\!\cdots\!40}{42\!\cdots\!21}a^{18}-\frac{12\!\cdots\!41}{42\!\cdots\!21}a^{17}+\frac{15\!\cdots\!83}{42\!\cdots\!21}a^{16}+\frac{19\!\cdots\!28}{42\!\cdots\!21}a^{15}+\frac{20\!\cdots\!25}{42\!\cdots\!21}a^{14}-\frac{19\!\cdots\!26}{42\!\cdots\!21}a^{13}-\frac{14\!\cdots\!94}{42\!\cdots\!21}a^{12}-\frac{11\!\cdots\!73}{42\!\cdots\!21}a^{11}+\frac{16\!\cdots\!35}{42\!\cdots\!21}a^{10}+\frac{72\!\cdots\!93}{42\!\cdots\!21}a^{9}-\frac{85\!\cdots\!80}{42\!\cdots\!21}a^{8}-\frac{11\!\cdots\!50}{42\!\cdots\!21}a^{7}+\frac{18\!\cdots\!98}{42\!\cdots\!21}a^{6}+\frac{19\!\cdots\!82}{42\!\cdots\!21}a^{5}-\frac{13\!\cdots\!46}{42\!\cdots\!21}a^{4}-\frac{55\!\cdots\!00}{42\!\cdots\!21}a^{3}-\frac{25\!\cdots\!52}{62\!\cdots\!87}a^{2}-\frac{17\!\cdots\!19}{42\!\cdots\!21}a-\frac{90\!\cdots\!27}{42\!\cdots\!21}$
| 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
| A cyclic group of order 45 |
| The 45 conjugacy class representatives for $C_{45}$ |
| Character table for $C_{45}$ |
Intermediate fields
| 3.3.361.1, 5.5.923521.1, 9.9.15072974715383053921.1, 15.15.4829212716211581952447142935561.1 |
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 | $45$ | $45$ | ${\href{/padicField/5.9.0.1}{9} }^{5}$ | ${\href{/padicField/7.5.0.1}{5} }^{9}$ | ${\href{/padicField/11.5.0.1}{5} }^{9}$ | $45$ | $45$ | R | $45$ | $45$ | R | ${\href{/padicField/37.3.0.1}{3} }^{15}$ | $45$ | $45$ | $45$ | $45$ | $45$ |
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 $45$ | $9$ | $5$ | $40$ | |||
|
\(31\)
| 31.15.14.6 | $x^{15} + 155$ | $15$ | $1$ | $14$ | $C_{15}$ | $[\ ]_{15}$ |
| 31.15.14.6 | $x^{15} + 155$ | $15$ | $1$ | $14$ | $C_{15}$ | $[\ ]_{15}$ | |
| 31.15.14.6 | $x^{15} + 155$ | $15$ | $1$ | $14$ | $C_{15}$ | $[\ ]_{15}$ |