Normalized defining polynomial
\( x^{24} - 23 x^{22} + 230 x^{20} - 1311 x^{18} + 4692 x^{16} - 10949 x^{14} + 16757 x^{12} - 16511 x^{10} + 10032 x^{8} - 3498 x^{6} + 628 x^{4} - 48 x^{2} + 1 \)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[24, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(5106705043047168064000000000000000000\) \(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{20}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(33.85\) | 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))
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Galois root discriminant: | $2\cdot 5^{3/4}7^{5/6}\approx 33.8458843070916$ | ||
Ramified primes: | \(2\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(140=2^{2}\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{140}(1,·)$, $\chi_{140}(131,·)$, $\chi_{140}(97,·)$, $\chi_{140}(9,·)$, $\chi_{140}(139,·)$, $\chi_{140}(13,·)$, $\chi_{140}(17,·)$, $\chi_{140}(19,·)$, $\chi_{140}(23,·)$, $\chi_{140}(29,·)$, $\chi_{140}(31,·)$, $\chi_{140}(107,·)$, $\chi_{140}(33,·)$, $\chi_{140}(67,·)$, $\chi_{140}(123,·)$, $\chi_{140}(81,·)$, $\chi_{140}(43,·)$, $\chi_{140}(109,·)$, $\chi_{140}(111,·)$, $\chi_{140}(117,·)$, $\chi_{140}(73,·)$, $\chi_{140}(121,·)$, $\chi_{140}(59,·)$, $\chi_{140}(127,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $23$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-3$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+3$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $2a^{23}-46a^{21}+459a^{19}-2602a^{17}+9215a^{15}-21113a^{13}+31330a^{11}-29304a^{9}+16268a^{7}-4811a^{5}+626a^{3}-23a$, $a$, $2a^{23}-46a^{21}+459a^{19}-2602a^{17}+9214a^{15}-21099a^{13}+31253a^{11}-29094a^{9}+15974a^{7}-4615a^{5}+577a^{3}-20a$, $a^{3}-3a$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{23}-23a^{21}+229a^{19}-1292a^{17}+4540a^{15}-10283a^{13}+15015a^{11}-13729a^{9}+7368a^{7}-2061a^{5}+246a^{3}-8a$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+a^{7}+294a^{6}-7a^{5}-196a^{4}+14a^{3}+49a^{2}-7a-2$, $2a^{23}-45a^{21}+439a^{19}-a^{18}-2433a^{17}+18a^{16}+8431a^{15}-135a^{14}-18943a^{13}+546a^{12}+27689a^{11}-1288a^{10}-25718a^{9}+1792a^{8}+14377a^{7}-1421a^{6}-4377a^{5}+590a^{4}+601a^{3}-106a^{2}-21a+4$, $a^{6}-6a^{4}+9a^{2}+a-2$, $2a^{23}+a^{22}-45a^{21}-22a^{20}+439a^{19}+210a^{18}-2432a^{17}-1140a^{16}+8414a^{15}+3874a^{14}-18824a^{13}-8541a^{12}+27247a^{11}+12232a^{10}-24783a^{9}-11055a^{8}+13255a^{7}+5896a^{6}-3663a^{5}-1629a^{4}+398a^{3}+181a^{2}-7a-5$, $a^{20}+a^{19}-20a^{18}-19a^{17}+169a^{16}+152a^{15}-784a^{14}-665a^{13}+2170a^{12}+1729a^{11}-3641a^{10}-2717a^{9}+3586a^{8}+2508a^{7}-1891a^{6}-1254a^{5}+434a^{4}+285a^{3}-24a^{2}-19a-1$, $a^{23}-23a^{21}-a^{20}+229a^{19}+19a^{18}-1292a^{17}-151a^{16}+4540a^{15}+650a^{14}-10283a^{13}-1638a^{12}+15016a^{11}+2431a^{10}-13740a^{9}-2014a^{8}+7411a^{7}+799a^{6}-2131a^{5}-90a^{4}+287a^{3}-8a^{2}-12a+1$, $2a^{23}-46a^{21}-a^{20}+459a^{19}+20a^{18}-2602a^{17}-170a^{16}+9215a^{15}+800a^{14}-21113a^{13}-2275a^{12}+31330a^{11}+4004a^{10}-29304a^{9}-4290a^{8}+16268a^{7}+2640a^{6}-4811a^{5}-825a^{4}+626a^{3}+100a^{2}-23a-2$, $2a^{23}-46a^{21}+459a^{19}-2602a^{17}+9215a^{15}-a^{14}-21113a^{13}+14a^{12}+31330a^{11}-77a^{10}-29304a^{9}+210a^{8}+16268a^{7}-294a^{6}-4811a^{5}+196a^{4}+626a^{3}-49a^{2}-23a+3$, $a^{22}-22a^{20}-a^{19}+210a^{18}+19a^{17}-1140a^{16}-152a^{15}+3874a^{14}+666a^{13}-8541a^{12}-1742a^{11}+12233a^{10}+2782a^{9}-11065a^{8}-2664a^{7}+5931a^{6}+1436a^{5}-1679a^{4}-376a^{3}+206a^{2}+32a-8$, $a^{22}-23a^{20}+229a^{18}-a^{17}-1292a^{16}+17a^{15}+4540a^{14}-119a^{13}-10283a^{12}+442a^{11}+15015a^{10}-935a^{9}-13729a^{8}+1122a^{7}+7368a^{6}-714a^{5}-2061a^{4}+204a^{3}+246a^{2}-17a-9$, $2a^{23}-46a^{21}+460a^{19}-2621a^{17}+9366a^{15}-21764a^{13}+32982a^{11}-31811a^{9}+18482a^{7}-5869a^{5}+862a^{3}-39a+1$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4005a^{10}-a^{9}+4300a^{8}+9a^{7}-2675a^{6}-27a^{5}+875a^{4}+30a^{3}-125a^{2}-9a+5$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 33400075541.82508 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{24}\cdot(2\pi)^{0}\cdot 33400075541.82508 \cdot 1}{2\cdot\sqrt{5106705043047168064000000000000000000}}\cr\approx \mathstrut & 0.123984376455386 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{12}$ (as 24T2):
An abelian group of order 24 |
The 24 conjugacy class representatives for $C_2\times C_{12}$ |
Character table for $C_2\times C_{12}$ is not computed |
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 | R | ${\href{/padicField/3.12.0.1}{12} }^{2}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{8}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $24$ | $2$ | $12$ | $24$ | |||
\(5\) | Deg $24$ | $4$ | $6$ | $18$ | |||
\(7\) | Deg $24$ | $6$ | $4$ | $20$ |