Normalized defining polynomial
\( x^{24} - 3 x^{22} + 8 x^{20} - 21 x^{18} + 55 x^{16} - 144 x^{14} + 377 x^{12} - 144 x^{10} + 55 x^{8} + \cdots + 1 \)
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: | \(326829122755018756096000000000000\) \(\medspace = 2^{24}\cdot 5^{12}\cdot 7^{20}\) | 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: | \(22.63\) | 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: | $2\cdot 5^{1/2}7^{5/6}\approx 22.634106993721137$ | ||
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$$\chi_{140}(1,·)$, $\chi_{140}(131,·)$, $\chi_{140}(69,·)$, $\chi_{140}(129,·)$, $\chi_{140}(9,·)$, $\chi_{140}(11,·)$, $\chi_{140}(79,·)$, $\chi_{140}(81,·)$, $\chi_{140}(19,·)$, $\chi_{140}(139,·)$, $\chi_{140}(89,·)$, $\chi_{140}(29,·)$, $\chi_{140}(31,·)$, $\chi_{140}(99,·)$, $\chi_{140}(101,·)$, $\chi_{140}(39,·)$, $\chi_{140}(41,·)$, $\chi_{140}(71,·)$, $\chi_{140}(109,·)$, $\chi_{140}(111,·)$, $\chi_{140}(51,·)$, $\chi_{140}(121,·)$, $\chi_{140}(59,·)$, $\chi_{140}(61,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{377}a^{14}-\frac{144}{377}$, $\frac{1}{377}a^{15}-\frac{144}{377}a$, $\frac{1}{377}a^{16}-\frac{144}{377}a^{2}$, $\frac{1}{377}a^{17}-\frac{144}{377}a^{3}$, $\frac{1}{377}a^{18}-\frac{144}{377}a^{4}$, $\frac{1}{377}a^{19}-\frac{144}{377}a^{5}$, $\frac{1}{377}a^{20}-\frac{144}{377}a^{6}$, $\frac{1}{377}a^{21}-\frac{144}{377}a^{7}$, $\frac{1}{377}a^{22}-\frac{144}{377}a^{8}$, $\frac{1}{377}a^{23}-\frac{144}{377}a^{9}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{5}{377} a^{19} - \frac{4181}{377} a^{5} \) (order $28$) | 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: | $\frac{1}{377}a^{17}+\frac{987}{377}a^{3}$, $\frac{3}{377}a^{18}-\frac{1}{377}a^{16}+\frac{2584}{377}a^{4}-\frac{987}{377}a^{2}$, $\frac{21}{377}a^{22}+\frac{17711}{377}a^{8}+1$, $\frac{34}{377}a^{23}+\frac{28657}{377}a^{9}+1$, $\frac{34}{377}a^{23}-\frac{21}{377}a^{22}+\frac{28657}{377}a^{9}-\frac{17711}{377}a^{8}$, $\frac{2}{377}a^{17}+\frac{1597}{377}a^{3}-1$, $\frac{165}{377}a^{23}-\frac{440}{377}a^{21}+\frac{40}{13}a^{19}-\frac{3027}{377}a^{17}+\frac{7920}{377}a^{15}-55a^{13}+144a^{11}-\frac{3025}{377}a^{9}+\frac{1155}{377}a^{7}+\frac{129}{13}a^{5}-\frac{1432}{377}a^{3}-\frac{55}{377}a$, $\frac{267}{377}a^{23}-\frac{720}{377}a^{21}+\frac{1869}{377}a^{19}-\frac{4893}{377}a^{17}+\frac{12816}{377}a^{15}-89a^{13}+233a^{11}-\frac{4895}{377}a^{9}-\frac{4896}{377}a^{7}-\frac{712}{377}a^{5}+\frac{1864}{377}a^{3}-\frac{89}{377}a$, $\frac{178}{377}a^{23}-\frac{699}{377}a^{21}+\frac{1864}{377}a^{19}+\frac{5}{377}a^{18}-\frac{4893}{377}a^{17}+\frac{12815}{377}a^{15}-89a^{13}+233a^{11}-\frac{79920}{377}a^{9}+\frac{12815}{377}a^{7}-\frac{4893}{377}a^{5}+\frac{4181}{377}a^{4}+\frac{1864}{377}a^{3}-\frac{699}{377}a-1$, $\frac{165}{377}a^{22}+\frac{8}{377}a^{21}-\frac{440}{377}a^{20}+\frac{1153}{377}a^{18}-\frac{3025}{377}a^{16}-\frac{1}{377}a^{15}+\frac{7920}{377}a^{14}-55a^{12}+144a^{10}-\frac{3025}{377}a^{8}+\frac{6765}{377}a^{7}+\frac{1155}{377}a^{6}-\frac{2037}{377}a^{4}+\frac{165}{377}a^{2}-\frac{610}{377}a-\frac{55}{377}$, $\frac{1}{29}a^{21}+\frac{1}{29}a^{20}+\frac{8}{377}a^{19}+\frac{3}{377}a^{18}+\frac{842}{29}a^{7}+\frac{842}{29}a^{6}+\frac{6765}{377}a^{5}+\frac{2584}{377}a^{4}$ (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: | \( 12419221.75230148 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 12419221.75230148 \cdot 3}{28\cdot\sqrt{326829122755018756096000000000000}}\cr\approx \mathstrut & 0.278647821008879 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times C_6$ (as 24T3):
An abelian group of order 24 |
The 24 conjugacy class representatives for $C_2^2\times C_6$ |
Character table for $C_2^2\times C_6$ |
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 | ${\href{/padicField/3.6.0.1}{6} }^{4}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.1.0.1}{1} }^{24}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
\(5\) | 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(7\) | 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |