Normalized defining polynomial
\( x^{24} - x^{23} - 23 x^{22} + 23 x^{21} + 230 x^{20} - 229 x^{19} - 1312 x^{18} + 1294 x^{17} + 4709 x^{16} - 4573 x^{15} - 11067 x^{14} + 10506 x^{13} + 17187 x^{12} - 15810 x^{11} + \cdots + 1 \)
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: | \(161761786626698377317203521728515625\) \(\medspace = 3^{12}\cdot 5^{18}\cdot 7^{20}\) | 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: | \(29.31\) | 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: | $3^{1/2}5^{3/4}7^{5/6}\approx 29.3113956234904$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | 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: | \(105=3\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(2,·)$, $\chi_{105}(4,·)$, $\chi_{105}(8,·)$, $\chi_{105}(73,·)$, $\chi_{105}(13,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(82,·)$, $\chi_{105}(23,·)$, $\chi_{105}(89,·)$, $\chi_{105}(26,·)$, $\chi_{105}(92,·)$, $\chi_{105}(32,·)$, $\chi_{105}(97,·)$, $\chi_{105}(101,·)$, $\chi_{105}(103,·)$, $\chi_{105}(104,·)$, $\chi_{105}(41,·)$, $\chi_{105}(46,·)$, $\chi_{105}(52,·)$, $\chi_{105}(53,·)$, $\chi_{105}(59,·)$$\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: | $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{23}-22a^{21}+a^{20}+209a^{19}-18a^{18}-1122a^{17}+136a^{16}+3740a^{15}-561a^{14}-8007a^{13}+1377a^{12}+10999a^{11}-2057a^{10}-9383a^{9}+1835a^{8}+4598a^{7}-911a^{6}-1079a^{5}+206a^{4}+58a^{3}-7a^{2}+8a-1$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{20}-20a^{18}+170a^{16}-a^{15}-800a^{14}+15a^{13}+2275a^{12}-90a^{11}-4004a^{10}+275a^{9}+4290a^{8}-450a^{7}-2640a^{6}+378a^{5}+825a^{4}-140a^{3}-100a^{2}+15a+3$, $a^{6}-6a^{4}+9a^{2}+a-2$, $a^{23}-23a^{21}+a^{20}+231a^{19}-19a^{18}-1330a^{17}+153a^{16}+4845a^{15}-680a^{14}-11627a^{13}+1819a^{12}+18551a^{11}-2992a^{10}-19383a^{9}+2958a^{8}+12713a^{7}-1632a^{6}-4817a^{5}+424a^{4}+899a^{3}-31a^{2}-58a-1$, $a^{23}-23a^{21}+a^{20}+231a^{19}-19a^{18}-1330a^{17}+153a^{16}+4845a^{15}-680a^{14}-11627a^{13}+1819a^{12}+18551a^{11}-2992a^{10}-19383a^{9}+2958a^{8}+12713a^{7}-1633a^{6}-4817a^{5}+430a^{4}+899a^{3}-40a^{2}-59a+1$, $a^{5}-5a^{3}+5a+1$, $2a^{23}-a^{22}-46a^{21}+22a^{20}+460a^{19}-207a^{18}-2624a^{17}+1087a^{16}+9418a^{15}-3486a^{14}-22134a^{13}+7019a^{12}+34373a^{11}-8778a^{10}-34771a^{9}+6489a^{8}+22023a^{7}-2534a^{6}-8091a^{5}+391a^{4}+1493a^{3}+4a^{2}-100a-2$, $a^{23}-23a^{21}+231a^{19}+a^{18}-1331a^{17}-17a^{16}+4861a^{15}+120a^{14}-11732a^{13}-457a^{12}+18916a^{11}+1024a^{10}-20108a^{9}-1386a^{8}+13541a^{7}+1120a^{6}-5334a^{5}-506a^{4}+1050a^{3}+106a^{2}-71a-6$, $a^{23}+a^{22}-22a^{21}-21a^{20}+211a^{19}+191a^{18}-1158a^{17}-986a^{16}+4013a^{15}+3180a^{14}-9143a^{13}-6643a^{12}+13831a^{11}+9008a^{10}-13719a^{9}-7714a^{8}+8616a^{7}+3904a^{6}-3209a^{5}-1016a^{4}+624a^{3}+96a^{2}-45a$, $a^{23}-23a^{21}+a^{20}+231a^{19}-19a^{18}-1329a^{17}+153a^{16}+4828a^{15}-680a^{14}-11508a^{13}+1820a^{12}+18109a^{11}-3004a^{10}-18448a^{9}+3012a^{8}+11591a^{7}-1745a^{6}-4103a^{5}+535a^{4}+695a^{3}-77a^{2}-42a+5$, $a^{23}+a^{22}-22a^{21}-21a^{20}+210a^{19}+190a^{18}-1140a^{17}-969a^{16}+3876a^{15}+3060a^{14}-8568a^{13}-6189a^{12}+12376a^{11}+8019a^{10}-11441a^{9}-6481a^{8}+6443a^{7}+3094a^{6}-2024a^{5}-800a^{4}+310a^{3}+96a^{2}-21a-2$, $a^{16}-16a^{14}+104a^{12}+a^{11}-352a^{10}-11a^{9}+660a^{8}+44a^{7}-672a^{6}-77a^{5}+335a^{4}+55a^{3}-60a^{2}-12a$, $a^{23}-a^{22}-23a^{21}+22a^{20}+229a^{19}-208a^{18}-1294a^{17}+1105a^{16}+4573a^{15}-3621a^{14}-10506a^{13}+7565a^{12}+15809a^{11}-10065a^{10}-15322a^{9}+8271a^{8}+9145a^{7}-3920a^{6}-3065a^{5}+932a^{4}+473a^{3}-80a^{2}-21a+1$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a$, $2a^{23}-45a^{21}+a^{20}+440a^{19}-18a^{18}-2452a^{17}+136a^{16}+8585a^{15}-561a^{14}-19634a^{13}+1377a^{12}+29549a^{11}-2057a^{10}-28756a^{9}+1835a^{8}+17276a^{7}-911a^{6}-5846a^{5}+206a^{4}+933a^{3}-8a^{2}-50a-1$, $a^{23}-23a^{21}+a^{20}+231a^{19}-19a^{18}-1329a^{17}+153a^{16}+4828a^{15}-680a^{14}-11508a^{13}+1819a^{12}+18109a^{11}-2992a^{10}-18448a^{9}+2957a^{8}+11591a^{7}-1625a^{6}-4103a^{5}+410a^{4}+694a^{3}-24a^{2}-39a-1$, $a^{20}-a^{19}-19a^{18}+18a^{17}+153a^{16}-136a^{15}-680a^{14}+561a^{13}+1819a^{12}-1376a^{11}-2993a^{10}+2047a^{9}+2967a^{8}-1801a^{7}-1659a^{6}+868a^{5}+455a^{4}-196a^{3}-43a^{2}+14a$, $a^{19}-18a^{17}+135a^{15}-546a^{13}+1287a^{11}-1782a^{9}+a^{8}+1386a^{7}-8a^{6}-540a^{5}+19a^{4}+81a^{3}-13a^{2}-2a+2$, $2a^{23}+a^{22}-44a^{21}-21a^{20}+419a^{19}+190a^{18}-2262a^{17}-968a^{16}+7616a^{15}+3044a^{14}-16576a^{13}-6085a^{12}+23387a^{11}+7667a^{10}-20878a^{9}-5821a^{8}+11153a^{7}+2421a^{6}-3208a^{5}-459a^{4}+405a^{3}+27a^{2}-16a$, $2a^{23}-46a^{21}+a^{20}+461a^{19}-18a^{18}-2642a^{17}+136a^{16}+9554a^{15}-561a^{14}-22695a^{13}+1377a^{12}+35751a^{11}-2058a^{10}-36839a^{9}+1845a^{8}+23903a^{7}-946a^{6}-9086a^{5}+256a^{4}+1768a^{3}-32a^{2}-127a+1$, $2a^{23}-46a^{21}+a^{20}+461a^{19}-18a^{18}-2642a^{17}+136a^{16}+9554a^{15}-561a^{14}-22694a^{13}+1376a^{12}+35737a^{11}-2045a^{10}-36762a^{9}+1780a^{8}+23693a^{7}-791a^{6}-8794a^{5}+81a^{4}+1583a^{3}+44a^{2}-93a-3$ (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: | \( 5459008653.485176 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{24}\cdot(2\pi)^{0}\cdot 5459008653.485176 \cdot 1}{2\cdot\sqrt{161761786626698377317203521728515625}}\cr\approx \mathstrut & 0.113858567512639 \end{aligned}\] (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 | ${\href{/padicField/2.12.0.1}{12} }^{2}$ | R | R | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $24$ | $2$ | $12$ | $12$ | |||
\(5\) | 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
\(7\) | Deg $24$ | $6$ | $4$ | $20$ |