Normalized defining polynomial
\( x^{24} - 24 x^{22} + 252 x^{20} - 1520 x^{18} + 5813 x^{16} - 14672 x^{14} + 24648 x^{12} - 27104 x^{10} + \cdots + 1 \)
Invariants
| Degree: | $24$ |
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| Signature: | $[24, 0]$ |
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| Discriminant: |
\(376808323956052112639025409344139165696\)
\(\medspace = 2^{72}\cdot 7^{20}\)
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| Root discriminant: | \(40.49\) |
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| Galois root discriminant: | $2^{3}7^{5/6}\approx 40.48912147837109$ | ||
| Ramified primes: |
\(2\), \(7\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_{12}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(112=2^{4}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{112}(1,·)$, $\chi_{112}(3,·)$, $\chi_{112}(65,·)$, $\chi_{112}(9,·)$, $\chi_{112}(75,·)$, $\chi_{112}(81,·)$, $\chi_{112}(19,·)$, $\chi_{112}(85,·)$, $\chi_{112}(87,·)$, $\chi_{112}(25,·)$, $\chi_{112}(27,·)$, $\chi_{112}(29,·)$, $\chi_{112}(31,·)$, $\chi_{112}(37,·)$, $\chi_{112}(103,·)$, $\chi_{112}(109,·)$, $\chi_{112}(93,·)$, $\chi_{112}(47,·)$, $\chi_{112}(83,·)$, $\chi_{112}(53,·)$, $\chi_{112}(55,·)$, $\chi_{112}(111,·)$, $\chi_{112}(57,·)$, $\chi_{112}(59,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
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
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
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Unit group
| Rank: | $23$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+3$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{22}-22a^{20}+208a^{18}-1104a^{16}+3604a^{14}-7448a^{12}+9648a^{10}-7456a^{8}+3075a^{6}-530a^{4}+24a^{2}$, $a^{23}-23a^{21}+230a^{19}-1312a^{17}+4708a^{15}-11052a^{13}+17096a^{11}-17104a^{9}+10531a^{7}-3605a^{5}+554a^{3}-24a$, $a$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-3$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+a^{7}+294a^{6}-7a^{5}-196a^{4}+14a^{3}+49a^{2}-7a-1$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a+1$, $a^{21}-22a^{19}+208a^{17}-1104a^{15}+3604a^{13}-7448a^{11}+9648a^{9}-7456a^{7}+3075a^{5}-530a^{3}+24a+1$, $a^{8}-a^{7}-8a^{6}+7a^{5}+20a^{4}-14a^{3}-16a^{2}+7a+2$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}-a+2$, $a^{16}-a^{15}-16a^{14}+15a^{13}+104a^{12}-90a^{11}-352a^{10}+275a^{9}+660a^{8}-450a^{7}-672a^{6}+378a^{5}+336a^{4}-140a^{3}-64a^{2}+15a+2$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}-a^{8}+9867a^{7}+8a^{6}-3289a^{5}-20a^{4}+506a^{3}+16a^{2}-23a-2$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a+1$, $a+1$, $a^{7}-7a^{5}+14a^{3}-7a+1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a+1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a-1$
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| Regulator: | \( 360688478301.3431 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{24}\cdot(2\pi)^{0}\cdot 360688478301.3431 \cdot 1}{2\cdot\sqrt{376808323956052112639025409344139165696}}\cr\approx \mathstrut & 0.155869781218936 \end{aligned}\] (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 $C_2\times C_{12}$ |
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 | ${\href{/padicField/3.12.0.1}{12} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }^{2}$ | R | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{8}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{8}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $24$ | $8$ | $3$ | $72$ | |||
|
\(7\)
| 7.2.6.10a1.2 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 192996 x^{6} + 266328 x^{5} + 234495 x^{4} + 131220 x^{3} + 45198 x^{2} + 8748 x + 736$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $$[\ ]_{6}^{2}$$ |
| 7.2.6.10a1.2 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 192996 x^{6} + 266328 x^{5} + 234495 x^{4} + 131220 x^{3} + 45198 x^{2} + 8748 x + 736$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $$[\ ]_{6}^{2}$$ |