Normalized defining polynomial
\( x^{24} - 11 x^{23} + 53 x^{22} - 131 x^{21} + 131 x^{20} + 144 x^{19} - 585 x^{18} + 471 x^{17} + 846 x^{16} - 3226 x^{15} + 6073 x^{14} - 7616 x^{13} + 4239 x^{12} + 5146 x^{11} - 14303 x^{10} + 15630 x^{9} - 9886 x^{8} + 3829 x^{7} - 1007 x^{6} + 240 x^{5} - 15 x^{4} - 39 x^{3} + 19 x^{2} - 3 x + 1 \)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[4, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(19756778413055716819205133752664064\)\(\medspace = 2^{16}\cdot 887^{10}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $26.85$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $2, 887$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $4$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{734875002403804819351514058255122} a^{23} + \frac{52527180720531312782089213442467}{367437501201902409675757029127561} a^{22} - \frac{128858027033476485067405523109275}{734875002403804819351514058255122} a^{21} + \frac{51933286546121821592948209032869}{734875002403804819351514058255122} a^{20} + \frac{53408547151287390963031269081425}{367437501201902409675757029127561} a^{19} - \frac{76455412351971200021086398659473}{367437501201902409675757029127561} a^{18} - \frac{12550891723131108778390767582313}{734875002403804819351514058255122} a^{17} - \frac{172766575632896549374670639343327}{734875002403804819351514058255122} a^{16} + \frac{81715108034567303699652284244241}{734875002403804819351514058255122} a^{15} + \frac{148584816296962143661857253771603}{734875002403804819351514058255122} a^{14} - \frac{300113323466706870981422181899705}{734875002403804819351514058255122} a^{13} + \frac{237508640984049361645207776862291}{734875002403804819351514058255122} a^{12} + \frac{97755250964274970117835334585471}{734875002403804819351514058255122} a^{11} - \frac{101161976728521577006751080390701}{367437501201902409675757029127561} a^{10} - \frac{13544998300887236871843652243664}{367437501201902409675757029127561} a^{9} + \frac{60147841296112989148164441931749}{734875002403804819351514058255122} a^{8} - \frac{110066360940465980619837380374237}{734875002403804819351514058255122} a^{7} + \frac{56996181849411659469861744961901}{367437501201902409675757029127561} a^{6} + \frac{180855740564491382237066909130551}{734875002403804819351514058255122} a^{5} - \frac{57453034900841302362907770325379}{367437501201902409675757029127561} a^{4} - \frac{97256006948571726161369547965294}{367437501201902409675757029127561} a^{3} + \frac{30190398514476596717978349955624}{367437501201902409675757029127561} a^{2} + \frac{66861399103974620059379785711468}{367437501201902409675757029127561} a - \frac{42506773650306783785854255813850}{367437501201902409675757029127561}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 33313220.242250312 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
A non-solvable group of order 240 |
The 18 conjugacy class representatives for $C_4.A_5$ |
Character table for $C_4.A_5$ |
Intermediate fields
6.2.12588304.1, Deg 12 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 40 siblings: | data not computed |
Arithmetically equvalently sibling: | 24.4.19756778413055716819205133752664064.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{8}$ | $20{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/31.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ | $20{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | $20{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
$2$ | 2.12.8.1 | $x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ |
2.12.8.1 | $x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
887 | Data not computed |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.887.2t1.a.a | $1$ | $ 887 $ | \(\Q(\sqrt{-887}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.3548.120.b.a | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
2.3548.120.b.b | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
2.3548.120.b.c | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
2.3548.120.b.d | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
3.3147076.12t33.a.a | $3$ | $ 2^{2} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
3.3147076.12t33.a.b | $3$ | $ 2^{2} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
* | 3.3548.12t76.a.a | $3$ | $ 2^{2} \cdot 887 $ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
* | 3.3548.12t76.a.b | $3$ | $ 2^{2} \cdot 887 $ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
4.3147076.10t11.a.a | $4$ | $ 2^{2} \cdot 887^{2}$ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $0$ | |
4.3147076.5t4.a.a | $4$ | $ 2^{2} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $0$ | |
4.3147076.40t188.b.a | $4$ | $ 2^{2} \cdot 887^{2}$ | 24.4.19756778413055716819205133752664064.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
4.3147076.40t188.b.b | $4$ | $ 2^{2} \cdot 887^{2}$ | 24.4.19756778413055716819205133752664064.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
5.11165825648.12t75.a.a | $5$ | $ 2^{4} \cdot 887^{3}$ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $-1$ | |
* | 5.12588304.6t12.a.a | $5$ | $ 2^{4} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $1$ |
* | 6.11165825648.24t576.b.a | $6$ | $ 2^{4} \cdot 887^{3}$ | 24.4.19756778413055716819205133752664064.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ |
* | 6.11165825648.24t576.b.b | $6$ | $ 2^{4} \cdot 887^{3}$ | 24.4.19756778413055716819205133752664064.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ |