Normalized defining polynomial
\( x^{24} - 18x^{18} + 323x^{12} - 18x^{6} + 1 \)
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: | \(614787626176508399616000000000000\) \(\medspace = 2^{24}\cdot 3^{36}\cdot 5^{12}\) | 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: | \(23.24\) | 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: | $2\cdot 3^{3/2}5^{1/2}\approx 23.2379000772445$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(180=2^{2}\cdot 3^{2}\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{180}(1,·)$, $\chi_{180}(131,·)$, $\chi_{180}(71,·)$, $\chi_{180}(11,·)$, $\chi_{180}(79,·)$, $\chi_{180}(19,·)$, $\chi_{180}(139,·)$, $\chi_{180}(149,·)$, $\chi_{180}(151,·)$, $\chi_{180}(89,·)$, $\chi_{180}(91,·)$, $\chi_{180}(29,·)$, $\chi_{180}(31,·)$, $\chi_{180}(161,·)$, $\chi_{180}(101,·)$, $\chi_{180}(119,·)$, $\chi_{180}(41,·)$, $\chi_{180}(109,·)$, $\chi_{180}(49,·)$, $\chi_{180}(179,·)$, $\chi_{180}(169,·)$, $\chi_{180}(121,·)$, $\chi_{180}(59,·)$, $\chi_{180}(61,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8}a^{12}+\frac{3}{8}a^{6}+\frac{1}{8}$, $\frac{1}{8}a^{13}+\frac{3}{8}a^{7}+\frac{1}{8}a$, $\frac{1}{8}a^{14}+\frac{3}{8}a^{8}+\frac{1}{8}a^{2}$, $\frac{1}{8}a^{15}+\frac{3}{8}a^{9}+\frac{1}{8}a^{3}$, $\frac{1}{8}a^{16}+\frac{3}{8}a^{10}+\frac{1}{8}a^{4}$, $\frac{1}{8}a^{17}+\frac{3}{8}a^{11}+\frac{1}{8}a^{5}$, $\frac{1}{2584}a^{18}-\frac{987}{2584}$, $\frac{1}{2584}a^{19}-\frac{987}{2584}a$, $\frac{1}{2584}a^{20}-\frac{987}{2584}a^{2}$, $\frac{1}{2584}a^{21}-\frac{987}{2584}a^{3}$, $\frac{1}{2584}a^{22}-\frac{987}{2584}a^{4}$, $\frac{1}{2584}a^{23}-\frac{987}{2584}a^{5}$
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: | \( -\frac{233}{2584} a^{19} + \frac{13}{8} a^{13} - \frac{233}{8} a^{7} + \frac{2097}{1292} a \) (order $36$) | 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: | $\frac{1}{2584}a^{20}+\frac{4181}{2584}a^{2}$, $\frac{1}{2584}a^{20}+\frac{6765}{2584}a^{2}-1$, $\frac{495}{1292}a^{22}+\frac{1}{2584}a^{20}-\frac{55}{8}a^{16}+\frac{987}{8}a^{10}-\frac{55}{2584}a^{4}+\frac{6765}{2584}a^{2}$, $\frac{9}{38}a^{21}+\frac{18}{323}a^{18}-\frac{17}{4}a^{15}-a^{12}+\frac{305}{4}a^{9}+18a^{6}-\frac{1}{76}a^{3}-\frac{324}{323}$, $\frac{5}{2584}a^{23}+\frac{28657}{2584}a^{5}-1$, $\frac{801}{1292}a^{23}-\frac{3}{2584}a^{22}-\frac{89}{8}a^{17}+\frac{1597}{8}a^{11}-\frac{89}{2584}a^{5}-\frac{17711}{2584}a^{4}$, $\frac{987}{2584}a^{22}+\frac{1}{2584}a^{20}-\frac{18}{323}a^{18}-\frac{55}{8}a^{16}+a^{12}+\frac{987}{8}a^{10}-18a^{6}-\frac{8883}{1292}a^{4}+\frac{4181}{2584}a^{2}+\frac{1}{323}$, $\frac{1597}{2584}a^{23}-\frac{987}{2584}a^{21}+\frac{233}{2584}a^{19}-\frac{89}{8}a^{17}+\frac{55}{8}a^{15}-\frac{13}{8}a^{13}+\frac{1597}{8}a^{11}-\frac{987}{8}a^{9}+\frac{233}{8}a^{7}-\frac{14373}{1292}a^{5}+\frac{8883}{1292}a^{3}-\frac{2097}{1292}a$, $\frac{987}{2584}a^{23}-\frac{495}{1292}a^{22}+\frac{1}{2584}a^{21}+\frac{377}{2584}a^{20}-\frac{18}{323}a^{19}-\frac{55}{8}a^{17}+\frac{55}{8}a^{16}-\frac{21}{8}a^{14}+a^{13}+\frac{987}{8}a^{11}-\frac{987}{8}a^{10}+\frac{377}{8}a^{8}-18a^{7}-\frac{8883}{1292}a^{5}+\frac{55}{2584}a^{4}+\frac{6765}{2584}a^{3}-\frac{3393}{1292}a^{2}+\frac{1}{323}a$, $\frac{1}{2584}a^{20}+\frac{18}{323}a^{19}-\frac{18}{323}a^{18}-a^{13}+a^{12}+18a^{7}-18a^{6}+\frac{6765}{2584}a^{2}-\frac{324}{323}a+\frac{1}{323}$, $\frac{1597}{2584}a^{23}+\frac{1597}{2584}a^{22}+\frac{305}{1292}a^{21}-\frac{89}{8}a^{17}-\frac{89}{8}a^{16}-\frac{17}{4}a^{15}+\frac{1597}{8}a^{11}+\frac{1597}{8}a^{10}+\frac{305}{4}a^{9}-\frac{14373}{1292}a^{5}-\frac{14373}{1292}a^{4}-\frac{2745}{646}a^{3}$ (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: | \( 13636610.630449586 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 13636610.630449586 \cdot 3}{36\cdot\sqrt{614787626176508399616000000000000}}\cr\approx \mathstrut & 0.173508619931334 \end{aligned}\] (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 | R | R | ${\href{/padicField/7.6.0.1}{6} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{12}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
\(3\) | 3.12.18.82 | $x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$ | $6$ | $2$ | $18$ | $C_6\times C_2$ | $[2]_{2}^{2}$ |
3.12.18.82 | $x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$ | $6$ | $2$ | $18$ | $C_6\times C_2$ | $[2]_{2}^{2}$ | |
\(5\) | 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 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |