Normalized defining polynomial
\( x^{24} + 91x^{20} + 1391x^{16} + 2688x^{12} + 1287x^{8} + 182x^{4} + 1 \)
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: | \(2283749599146799148302336000000000000\) \(\medspace = 2^{48}\cdot 5^{12}\cdot 7^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.73\) | 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))
| |
Galois root discriminant: | $2^{2}5^{1/2}7^{2/3}\approx 32.72982527225799$ | ||
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: | \(280=2^{3}\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(261,·)$, $\chi_{280}(51,·)$, $\chi_{280}(71,·)$, $\chi_{280}(9,·)$, $\chi_{280}(11,·)$, $\chi_{280}(239,·)$, $\chi_{280}(141,·)$, $\chi_{280}(79,·)$, $\chi_{280}(81,·)$, $\chi_{280}(211,·)$, $\chi_{280}(149,·)$, $\chi_{280}(151,·)$, $\chi_{280}(219,·)$, $\chi_{280}(29,·)$, $\chi_{280}(99,·)$, $\chi_{280}(39,·)$, $\chi_{280}(169,·)$, $\chi_{280}(109,·)$, $\chi_{280}(221,·)$, $\chi_{280}(179,·)$, $\chi_{280}(121,·)$, $\chi_{280}(249,·)$, $\chi_{280}(191,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{8}-\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{10}-\frac{1}{3}a^{6}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{3}$, $\frac{1}{87}a^{16}-\frac{4}{29}a^{12}+\frac{10}{87}a^{8}+\frac{31}{87}a^{4}-\frac{4}{87}$, $\frac{1}{87}a^{17}-\frac{4}{29}a^{13}+\frac{10}{87}a^{9}+\frac{31}{87}a^{5}-\frac{4}{87}a$, $\frac{1}{87}a^{18}-\frac{4}{29}a^{14}+\frac{10}{87}a^{10}+\frac{31}{87}a^{6}-\frac{4}{87}a^{2}$, $\frac{1}{87}a^{19}-\frac{4}{29}a^{15}+\frac{10}{87}a^{11}+\frac{31}{87}a^{7}-\frac{4}{87}a^{3}$, $\frac{1}{11988687}a^{20}-\frac{44207}{11988687}a^{16}+\frac{544270}{11988687}a^{12}+\frac{2211407}{11988687}a^{8}+\frac{929836}{3996229}a^{4}-\frac{1401487}{11988687}$, $\frac{1}{11988687}a^{21}-\frac{44207}{11988687}a^{17}+\frac{544270}{11988687}a^{13}+\frac{2211407}{11988687}a^{9}+\frac{929836}{3996229}a^{5}-\frac{1401487}{11988687}a$, $\frac{1}{11988687}a^{22}-\frac{44207}{11988687}a^{18}+\frac{544270}{11988687}a^{14}+\frac{2211407}{11988687}a^{10}+\frac{929836}{3996229}a^{6}-\frac{1401487}{11988687}a^{2}$, $\frac{1}{11988687}a^{23}-\frac{44207}{11988687}a^{19}+\frac{544270}{11988687}a^{15}+\frac{2211407}{11988687}a^{11}+\frac{929836}{3996229}a^{7}-\frac{1401487}{11988687}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{375523}{11988687} a^{21} - \frac{34178056}{11988687} a^{17} - \frac{522828208}{11988687} a^{13} - \frac{1015070498}{11988687} a^{9} - \frac{156565458}{3996229} a^{5} - \frac{46242242}{11988687} a \) (order $8$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{1300859}{3996229}a^{23}+\frac{118548367}{3996229}a^{19}+\frac{1824920195}{3996229}a^{15}+\frac{3727776101}{3996229}a^{11}+\frac{2046894315}{3996229}a^{7}+\frac{322334222}{3996229}a^{3}$, $\frac{665846}{11988687}a^{22}+\frac{60836525}{11988687}a^{18}+\frac{316130941}{3996229}a^{14}+\frac{708403187}{3996229}a^{10}+\frac{1441531697}{11988687}a^{6}+\frac{97910943}{3996229}a^{2}$, $\frac{199640}{3996229}a^{22}+\frac{53956687}{11988687}a^{18}+\frac{783671744}{11988687}a^{14}+\frac{867353768}{11988687}a^{10}-\frac{460006726}{11988687}a^{6}-\frac{66756352}{3996229}a^{2}$, $\frac{30212}{137801}a^{22}+\frac{238228634}{11988687}a^{18}+\frac{3569113265}{11988687}a^{14}+\frac{1918743886}{3996229}a^{10}+\frac{398237995}{3996229}a^{6}-\frac{93018658}{11988687}a^{2}$, $\frac{1505218}{11988687}a^{22}+\frac{136873346}{11988687}a^{18}+\frac{2084519656}{11988687}a^{14}+\frac{1301545993}{3996229}a^{10}+\frac{1664184067}{11988687}a^{6}+\frac{177862147}{11988687}a^{2}$, $\frac{1253679}{3996229}a^{23}-\frac{6020}{97469}a^{22}+\frac{114069059}{3996229}a^{19}-\frac{1643771}{292407}a^{18}+\frac{5227289627}{11988687}a^{15}-\frac{8383274}{97469}a^{14}+\frac{10042358653}{11988687}a^{11}-\frac{48983498}{292407}a^{10}+\frac{4689687892}{11988687}a^{7}-\frac{24170447}{292407}a^{6}+\frac{602658088}{11988687}a^{3}-\frac{4001842}{292407}a^{2}+1$, $\frac{1253679}{3996229}a^{23}-\frac{375523}{11988687}a^{21}-\frac{129745}{3996229}a^{20}+\frac{114069059}{3996229}a^{19}-\frac{34178056}{11988687}a^{17}-\frac{11766893}{3996229}a^{16}+\frac{5227289627}{11988687}a^{15}-\frac{522828208}{11988687}a^{13}-\frac{176849582}{3996229}a^{12}+\frac{10042358653}{11988687}a^{11}-\frac{1015070498}{11988687}a^{9}-\frac{293748021}{3996229}a^{8}+\frac{4689687892}{11988687}a^{7}-\frac{156565458}{3996229}a^{5}-\frac{67833122}{3996229}a^{4}+\frac{602658088}{11988687}a^{3}-\frac{46242242}{11988687}a+\frac{6822310}{3996229}$, $\frac{6020}{97469}a^{22}-\frac{1654001}{11988687}a^{21}+\frac{1643771}{292407}a^{18}-\frac{50031297}{3996229}a^{17}+\frac{8383274}{97469}a^{14}-\frac{2262613313}{11988687}a^{13}+\frac{48983498}{292407}a^{10}-\frac{3873807392}{11988687}a^{9}+\frac{24170447}{292407}a^{6}-\frac{1185024337}{11988687}a^{5}+\frac{4001842}{292407}a^{2}-\frac{20336052}{3996229}a+1$, $\frac{1206499}{3996229}a^{23}+\frac{676508}{3996229}a^{22}-\frac{28106}{413403}a^{20}+\frac{109589751}{3996229}a^{19}+\frac{184271947}{11988687}a^{18}-\frac{850031}{137801}a^{16}+\frac{4979818669}{11988687}a^{15}+\frac{928480507}{3996229}a^{14}-\frac{12803363}{137801}a^{12}+\frac{8901389003}{11988687}a^{11}+\frac{4888877890}{11988687}a^{10}-\frac{65221975}{413403}a^{8}+\frac{3238692839}{11988687}a^{7}+\frac{1654720711}{11988687}a^{6}-\frac{6212011}{137801}a^{4}+\frac{238313510}{11988687}a^{3}+\frac{107250398}{11988687}a^{2}-\frac{360029}{413403}$, $\frac{3385514}{11988687}a^{22}-\frac{1115983}{11988687}a^{21}+\frac{328333}{11988687}a^{20}+\frac{308029121}{11988687}a^{18}-\frac{101572667}{11988687}a^{17}+\frac{9955072}{3996229}a^{16}+\frac{4704461419}{11988687}a^{14}-\frac{1553970910}{11988687}a^{13}+\frac{455519405}{11988687}a^{12}+\frac{9027288155}{11988687}a^{10}-\frac{1007797972}{3996229}a^{9}+\frac{288101257}{3996229}a^{8}+\frac{4219991518}{11988687}a^{6}-\frac{1460684701}{11988687}a^{5}+\frac{387845728}{11988687}a^{4}+\frac{556415846}{11988687}a^{2}-\frac{198329077}{11988687}a+\frac{42481205}{11988687}$, $\frac{7420264}{11988687}a^{23}-\frac{1654001}{11988687}a^{21}-\frac{284213}{3996229}a^{20}+\frac{674488891}{11988687}a^{19}-\frac{50031297}{3996229}a^{17}-\frac{77263817}{11988687}a^{16}+\frac{3417703403}{3996229}a^{15}-\frac{2262613313}{11988687}a^{13}-\frac{1156441270}{11988687}a^{12}+\frac{6305684264}{3996229}a^{11}-\frac{3873807392}{11988687}a^{9}-\frac{1848543682}{11988687}a^{8}+\frac{7847225527}{11988687}a^{7}-\frac{1185024337}{11988687}a^{5}-\frac{13047668}{413403}a^{4}+\frac{307068513}{3996229}a^{3}-\frac{24332281}{3996229}a+\frac{1384390}{3996229}$ (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)]
| |
Regulator: | \( 27409659.83250929 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 27409659.83250929 \cdot 27}{8\cdot\sqrt{2283749599146799148302336000000000000}}\cr\approx \mathstrut & 0.231745655457395 \end{aligned}\] (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 | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.1.0.1}{1} }^{24}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $24$ | $4$ | $6$ | $48$ | |||
\(5\) | 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 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(7\) | 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |