Normalized defining polynomial
\( x^{42} - 19 x^{41} + 67 x^{40} + 912 x^{39} - 7330 x^{38} - 9004 x^{37} + 242768 x^{36} - 354937 x^{35} + \cdots + 44461 \)
Invariants
| Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[42, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(538\!\cdots\!413\)
\(\medspace = 13^{21}\cdot 43^{40}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(129.62\) | 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: | $13^{1/2}43^{20/21}\approx 129.6151955955181$ | ||
| Ramified primes: |
\(13\), \(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(559=13\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{559}(1,·)$, $\chi_{559}(259,·)$, $\chi_{559}(391,·)$, $\chi_{559}(324,·)$, $\chi_{559}(14,·)$, $\chi_{559}(365,·)$, $\chi_{559}(272,·)$, $\chi_{559}(274,·)$, $\chi_{559}(532,·)$, $\chi_{559}(25,·)$, $\chi_{559}(547,·)$, $\chi_{559}(326,·)$, $\chi_{559}(38,·)$, $\chi_{559}(40,·)$, $\chi_{559}(298,·)$, $\chi_{559}(428,·)$, $\chi_{559}(53,·)$, $\chi_{559}(311,·)$, $\chi_{559}(441,·)$, $\chi_{559}(443,·)$, $\chi_{559}(181,·)$, $\chi_{559}(64,·)$, $\chi_{559}(66,·)$, $\chi_{559}(196,·)$, $\chi_{559}(454,·)$, $\chi_{559}(183,·)$, $\chi_{559}(79,·)$, $\chi_{559}(337,·)$, $\chi_{559}(339,·)$, $\chi_{559}(142,·)$, $\chi_{559}(90,·)$, $\chi_{559}(207,·)$, $\chi_{559}(92,·)$, $\chi_{559}(350,·)$, $\chi_{559}(144,·)$, $\chi_{559}(482,·)$, $\chi_{559}(103,·)$, $\chi_{559}(402,·)$, $\chi_{559}(246,·)$, $\chi_{559}(404,·)$, $\chi_{559}(508,·)$, $\chi_{559}(170,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{59775013457}a^{40}+\frac{3845516951}{59775013457}a^{39}+\frac{24582110822}{59775013457}a^{38}+\frac{17477751692}{59775013457}a^{37}+\frac{18621141935}{59775013457}a^{36}+\frac{2021997463}{59775013457}a^{35}-\frac{28575949981}{59775013457}a^{34}+\frac{23196676798}{59775013457}a^{33}-\frac{28860836890}{59775013457}a^{32}-\frac{12329561075}{59775013457}a^{31}+\frac{24830226425}{59775013457}a^{30}-\frac{29150613835}{59775013457}a^{29}-\frac{4255329489}{59775013457}a^{28}-\frac{16784453232}{59775013457}a^{27}-\frac{5490442125}{59775013457}a^{26}-\frac{21394202801}{59775013457}a^{25}+\frac{29167229584}{59775013457}a^{24}+\frac{23471829160}{59775013457}a^{23}-\frac{28403326286}{59775013457}a^{22}+\frac{5081802683}{59775013457}a^{21}+\frac{4191470145}{59775013457}a^{20}+\frac{19546845268}{59775013457}a^{19}-\frac{27526029838}{59775013457}a^{18}+\frac{20978766827}{59775013457}a^{17}+\frac{3479560153}{59775013457}a^{16}-\frac{14070163094}{59775013457}a^{15}-\frac{13775485550}{59775013457}a^{14}-\frac{10672725091}{59775013457}a^{13}-\frac{29328372407}{59775013457}a^{12}+\frac{14176900978}{59775013457}a^{11}-\frac{10857053946}{59775013457}a^{10}-\frac{23220625636}{59775013457}a^{9}+\frac{8143029247}{59775013457}a^{8}+\frac{16100542435}{59775013457}a^{7}+\frac{29446637782}{59775013457}a^{6}+\frac{18019114071}{59775013457}a^{5}+\frac{22322748817}{59775013457}a^{4}-\frac{22006793820}{59775013457}a^{3}+\frac{24907739021}{59775013457}a^{2}+\frac{24054892943}{59775013457}a+\frac{344396}{1344437}$, $\frac{1}{12\!\cdots\!33}a^{41}+\frac{74\!\cdots\!65}{12\!\cdots\!33}a^{40}+\frac{16\!\cdots\!85}{12\!\cdots\!33}a^{39}-\frac{12\!\cdots\!75}{12\!\cdots\!33}a^{38}+\frac{45\!\cdots\!54}{12\!\cdots\!33}a^{37}+\frac{30\!\cdots\!52}{12\!\cdots\!33}a^{36}+\frac{43\!\cdots\!37}{12\!\cdots\!33}a^{35}-\frac{61\!\cdots\!01}{12\!\cdots\!33}a^{34}+\frac{51\!\cdots\!02}{12\!\cdots\!33}a^{33}-\frac{58\!\cdots\!79}{12\!\cdots\!33}a^{32}+\frac{28\!\cdots\!33}{12\!\cdots\!33}a^{31}+\frac{36\!\cdots\!79}{12\!\cdots\!33}a^{30}-\frac{19\!\cdots\!28}{12\!\cdots\!33}a^{29}+\frac{60\!\cdots\!21}{12\!\cdots\!33}a^{28}-\frac{49\!\cdots\!74}{12\!\cdots\!33}a^{27}-\frac{11\!\cdots\!04}{12\!\cdots\!33}a^{26}+\frac{70\!\cdots\!98}{12\!\cdots\!33}a^{25}-\frac{18\!\cdots\!83}{12\!\cdots\!33}a^{24}-\frac{27\!\cdots\!47}{12\!\cdots\!33}a^{23}+\frac{46\!\cdots\!65}{12\!\cdots\!33}a^{22}+\frac{27\!\cdots\!07}{12\!\cdots\!33}a^{21}+\frac{31\!\cdots\!30}{12\!\cdots\!33}a^{20}+\frac{26\!\cdots\!79}{12\!\cdots\!33}a^{19}-\frac{27\!\cdots\!36}{12\!\cdots\!33}a^{18}+\frac{17\!\cdots\!14}{12\!\cdots\!33}a^{17}-\frac{33\!\cdots\!85}{12\!\cdots\!33}a^{16}+\frac{57\!\cdots\!63}{12\!\cdots\!33}a^{15}-\frac{38\!\cdots\!96}{12\!\cdots\!33}a^{14}+\frac{49\!\cdots\!00}{12\!\cdots\!33}a^{13}+\frac{55\!\cdots\!30}{12\!\cdots\!33}a^{12}-\frac{47\!\cdots\!25}{12\!\cdots\!33}a^{11}+\frac{36\!\cdots\!96}{12\!\cdots\!33}a^{10}-\frac{48\!\cdots\!48}{12\!\cdots\!33}a^{9}-\frac{50\!\cdots\!80}{12\!\cdots\!33}a^{8}+\frac{33\!\cdots\!21}{12\!\cdots\!33}a^{7}+\frac{46\!\cdots\!03}{12\!\cdots\!33}a^{6}-\frac{26\!\cdots\!30}{12\!\cdots\!33}a^{5}-\frac{50\!\cdots\!65}{12\!\cdots\!33}a^{4}-\frac{23\!\cdots\!94}{12\!\cdots\!33}a^{3}-\frac{29\!\cdots\!52}{12\!\cdots\!33}a^{2}+\frac{13\!\cdots\!54}{12\!\cdots\!33}a+\frac{10\!\cdots\!43}{28\!\cdots\!53}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $41$ | 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 42 |
| The 42 conjugacy class representatives for $C_{42}$ |
| Character table for $C_{42}$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 3.3.1849.1, 6.6.7511105797.1, 7.7.6321363049.1, 14.14.2507407572395754328761947317.1, \(\Q(\zeta_{43})^+\) |
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 | ${\href{/padicField/2.14.0.1}{14} }^{3}$ | $21^{2}$ | $42$ | ${\href{/padicField/7.6.0.1}{6} }^{7}$ | ${\href{/padicField/11.14.0.1}{14} }^{3}$ | R | $21^{2}$ | $42$ | $21^{2}$ | $21^{2}$ | $42$ | ${\href{/padicField/37.6.0.1}{6} }^{7}$ | ${\href{/padicField/41.14.0.1}{14} }^{3}$ | R | ${\href{/padicField/47.14.0.1}{14} }^{3}$ | $21^{2}$ | ${\href{/padicField/59.14.0.1}{14} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| Deg $42$ | $2$ | $21$ | $21$ | |||
|
\(43\)
| 43.21.20.1 | $x^{21} + 43$ | $21$ | $1$ | $20$ | $C_{21}$ | $[\ ]_{21}$ |
| 43.21.20.1 | $x^{21} + 43$ | $21$ | $1$ | $20$ | $C_{21}$ | $[\ ]_{21}$ |