Normalized defining polynomial
\( x^{24} + 25 x^{22} + 274 x^{20} + 1729 x^{18} + 6936 x^{16} + 18427 x^{14} + 32745 x^{12} + 38377 x^{10} + \cdots + 1 \)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 12]$ | 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)))
| |
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$$\chi_{140}(1,·)$, $\chi_{140}(3,·)$, $\chi_{140}(97,·)$, $\chi_{140}(81,·)$, $\chi_{140}(71,·)$, $\chi_{140}(9,·)$, $\chi_{140}(11,·)$, $\chi_{140}(13,·)$, $\chi_{140}(79,·)$, $\chi_{140}(17,·)$, $\chi_{140}(83,·)$, $\chi_{140}(87,·)$, $\chi_{140}(27,·)$, $\chi_{140}(29,·)$, $\chi_{140}(33,·)$, $\chi_{140}(99,·)$, $\chi_{140}(39,·)$, $\chi_{140}(103,·)$, $\chi_{140}(109,·)$, $\chi_{140}(47,·)$, $\chi_{140}(51,·)$, $\chi_{140}(117,·)$, $\chi_{140}(73,·)$, $\chi_{140}(121,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
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
$C_{78}$, which has order $78$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -a^{21} - 21 a^{19} - 189 a^{17} - 952 a^{15} - 2940 a^{13} - 5733 a^{11} - 7007 a^{9} - 5147 a^{7} - 2072 a^{5} - 371 a^{3} - 14 a \) (order $4$) | 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: | $a^{7}+7a^{5}+14a^{3}+7a$, $a^{10}+10a^{8}+35a^{6}+50a^{4}+25a^{2}+1$, $a^{10}+10a^{8}+35a^{6}+50a^{4}+25a^{2}+2$, $a^{15}+15a^{13}+90a^{11}+275a^{9}+449a^{7}+370a^{5}+121a^{3}+3a$, $a^{15}+15a^{13}+90a^{11}+275a^{9}+449a^{7}+371a^{5}+126a^{3}+8a$, $a^{19}+20a^{17}+170a^{15}+799a^{13}+2261a^{11}+3926a^{9}+4070a^{7}+2311a^{5}+579a^{3}+26a$, $a^{22}+22a^{20}+210a^{18}+1140a^{16}+3874a^{14}+8539a^{12}+12209a^{10}+10955a^{8}+5691a^{6}+1429a^{4}+102a^{2}+1$, $a^{17}+17a^{15}+119a^{13}+442a^{11}+935a^{9}+1121a^{7}+707a^{5}+190a^{3}+10a$, $a^{22}+21a^{20}+190a^{18}+970a^{16}+3074a^{14}+6264a^{12}+8205a^{10}+6665a^{8}+3051a^{6}+604a^{4}+2a^{2}-1$, $a^{17}+17a^{15}+119a^{13}+442a^{11}+935a^{9}+1122a^{7}+713a^{5}+199a^{3}+12a$, $a^{14}+14a^{12}+77a^{10}+210a^{8}+294a^{6}+195a^{4}+44a^{2}-3$ (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: | \( 3391665.6012423597 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 3391665.6012423597 \cdot 78}{4\cdot\sqrt{5106705043047168064000000000000000000}}\cr\approx \mathstrut & 0.110798956763764 \end{aligned}\] (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 | ${\href{/padicField/3.12.0.1}{12} }^{2}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 $24$ | $2$ | $12$ | $24$ | |||
\(5\) | 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
\(7\) | 7.12.10.5 | $x^{12} - 154 x^{6} - 1421$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ |
7.12.10.5 | $x^{12} - 154 x^{6} - 1421$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ |