Normalized defining polynomial
\( x^{24} + 4 x^{22} + 14 x^{20} + 48 x^{18} + 164 x^{16} + 560 x^{14} + 1912 x^{12} + 1120 x^{10} + \cdots + 64 \)
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: | \(5887630061813314259984772021002174464\) \(\medspace = 2^{66}\cdot 7^{20}\) | 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: | \(34.05\) | 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: | $2^{11/4}7^{5/6}\approx 34.04715710793443$ | ||
Ramified primes: | \(2\), \(7\) | 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: | \(112=2^{4}\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{112}(1,·)$, $\chi_{112}(97,·)$, $\chi_{112}(5,·)$, $\chi_{112}(65,·)$, $\chi_{112}(9,·)$, $\chi_{112}(13,·)$, $\chi_{112}(109,·)$, $\chi_{112}(17,·)$, $\chi_{112}(85,·)$, $\chi_{112}(89,·)$, $\chi_{112}(25,·)$, $\chi_{112}(101,·)$, $\chi_{112}(29,·)$, $\chi_{112}(69,·)$, $\chi_{112}(33,·)$, $\chi_{112}(37,·)$, $\chi_{112}(81,·)$, $\chi_{112}(41,·)$, $\chi_{112}(45,·)$, $\chi_{112}(93,·)$, $\chi_{112}(53,·)$, $\chi_{112}(73,·)$, $\chi_{112}(57,·)$, $\chi_{112}(61,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{1912}a^{14}-\frac{99}{239}$, $\frac{1}{1912}a^{15}-\frac{99}{239}a$, $\frac{1}{3824}a^{16}+\frac{70}{239}a^{2}$, $\frac{1}{3824}a^{17}+\frac{70}{239}a^{3}$, $\frac{1}{3824}a^{18}-\frac{99}{478}a^{4}$, $\frac{1}{3824}a^{19}-\frac{99}{478}a^{5}$, $\frac{1}{7648}a^{20}+\frac{35}{239}a^{6}$, $\frac{1}{7648}a^{21}+\frac{35}{239}a^{7}$, $\frac{1}{7648}a^{22}-\frac{99}{956}a^{8}$, $\frac{1}{7648}a^{23}-\frac{99}{956}a^{9}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{52}$, which has order $52$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{41}{3824} a^{22} - \frac{287}{7648} a^{20} - \frac{123}{956} a^{18} - \frac{1681}{3824} a^{16} - \frac{1435}{956} a^{14} - \frac{41}{8} a^{12} - \frac{35}{2} a^{10} - \frac{1681}{956} a^{8} - \frac{246}{239} a^{6} - \frac{287}{478} a^{4} - \frac{82}{239} a^{2} - \frac{41}{239} \) (order $14$) | 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: | $\frac{7}{7648}a^{20}+\frac{1}{3824}a^{16}-\frac{1189}{239}a^{6}-\frac{408}{239}a^{2}$, $\frac{3}{1912}a^{22}-\frac{8119}{956}a^{8}-1$, $\frac{3}{3824}a^{16}-\frac{985}{239}a^{2}$, $\frac{35}{3824}a^{23}+\frac{35}{956}a^{21}-\frac{7}{7648}a^{20}+\frac{245}{1912}a^{19}+\frac{105}{239}a^{17}+\frac{1435}{956}a^{15}+\frac{41}{8}a^{13}+\frac{35}{2}a^{11}+\frac{2450}{239}a^{9}+\frac{1435}{239}a^{7}+\frac{1189}{239}a^{6}+\frac{840}{239}a^{5}+\frac{490}{239}a^{3}+\frac{280}{239}a$, $\frac{3}{1912}a^{22}-\frac{1}{1912}a^{15}-\frac{8119}{956}a^{8}+\frac{577}{239}a$, $\frac{3}{1912}a^{22}+\frac{7}{7648}a^{20}-\frac{1}{3824}a^{18}+\frac{1}{3824}a^{16}-\frac{8119}{956}a^{8}-\frac{1189}{239}a^{6}+\frac{577}{478}a^{4}-\frac{408}{239}a^{2}-1$, $\frac{7}{7648}a^{20}+\frac{3}{3824}a^{18}+\frac{1}{3824}a^{16}-\frac{1189}{239}a^{6}-\frac{985}{239}a^{4}-\frac{408}{239}a^{2}$, $\frac{3}{1912}a^{23}+\frac{1}{1912}a^{18}-\frac{8119}{956}a^{9}-\frac{1393}{478}a^{4}-1$, $\frac{169}{7648}a^{23}+\frac{35}{3824}a^{22}+\frac{169}{1912}a^{21}+\frac{287}{7648}a^{20}+\frac{297}{956}a^{19}+\frac{123}{956}a^{18}+\frac{4059}{3824}a^{17}+\frac{1681}{3824}a^{16}+\frac{3465}{956}a^{15}+\frac{1435}{956}a^{14}+\frac{99}{8}a^{13}+\frac{41}{8}a^{12}+\frac{169}{4}a^{11}+\frac{35}{2}a^{10}+\frac{5915}{239}a^{9}+\frac{2450}{239}a^{8}+\frac{6929}{478}a^{7}+\frac{246}{239}a^{6}+\frac{693}{478}a^{5}+\frac{287}{478}a^{4}+\frac{198}{239}a^{3}+\frac{82}{239}a^{2}+\frac{99}{239}a+\frac{41}{239}$, $\frac{29}{7648}a^{23}+\frac{99}{3824}a^{22}+\frac{693}{7648}a^{20}+\frac{5}{3824}a^{19}+\frac{297}{956}a^{18}+\frac{4059}{3824}a^{16}+\frac{1}{1912}a^{15}+\frac{6929}{1912}a^{14}+\frac{99}{8}a^{12}+\frac{169}{4}a^{10}-\frac{19601}{956}a^{9}+\frac{4059}{956}a^{8}+\frac{594}{239}a^{6}-\frac{3363}{478}a^{5}+\frac{693}{478}a^{4}+\frac{198}{239}a^{2}-\frac{577}{239}a+\frac{676}{239}$, $\frac{3}{1912}a^{23}+\frac{35}{3824}a^{22}+\frac{287}{7648}a^{20}+\frac{245}{1912}a^{18}+\frac{105}{239}a^{16}+\frac{1435}{956}a^{14}+\frac{41}{8}a^{12}+\frac{35}{2}a^{10}-\frac{8119}{956}a^{9}+\frac{2450}{239}a^{8}+\frac{246}{239}a^{6}+\frac{840}{239}a^{4}+\frac{490}{239}a^{2}+\frac{280}{239}$ (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: | \( 39019312.21180779 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 39019312.21180779 \cdot 52}{14\cdot\sqrt{5887630061813314259984772021002174464}}\cr\approx \mathstrut & 0.226122136950489 \end{aligned}\] (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 | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.33.375 | $x^{12} + 80 x^{10} - 264 x^{9} - 638 x^{8} + 64 x^{7} + 208 x^{6} + 3904 x^{5} + 8348 x^{4} + 10496 x^{3} + 13152 x^{2} + 7200 x + 6392$ | $4$ | $3$ | $33$ | $C_{12}$ | $[3, 4]^{3}$ |
2.12.33.375 | $x^{12} + 80 x^{10} - 264 x^{9} - 638 x^{8} + 64 x^{7} + 208 x^{6} + 3904 x^{5} + 8348 x^{4} + 10496 x^{3} + 13152 x^{2} + 7200 x + 6392$ | $4$ | $3$ | $33$ | $C_{12}$ | $[3, 4]^{3}$ | |
\(7\) | 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 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |