Normalized defining polynomial
\( x^{24} - 3 x^{23} + 6 x^{22} - 19 x^{21} + 39 x^{20} - 30 x^{19} - 9 x^{18} - 45 x^{17} + 93 x^{16} + 204 x^{15} - 201 x^{14} - 129 x^{13} - 793 x^{12} + 738 x^{11} + 141 x^{10} + 1177 x^{9} - 666 x^{8} - 597 x^{7} - 913 x^{6} + 168 x^{5} + 474 x^{4} + 464 x^{3} + 168 x^{2} + 24 x - 4 \)
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: | \(7654476884289954721505578121232384\)\(\medspace = 2^{16}\cdot 3^{26}\cdot 11^{16}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $25.81$ | 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: | $2, 3, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{18} + \frac{1}{12} a^{15} - \frac{1}{4} a^{14} + \frac{1}{12} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{12} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{19} + \frac{1}{24} a^{16} + \frac{1}{8} a^{15} - \frac{1}{4} a^{14} + \frac{1}{24} a^{13} + \frac{1}{8} a^{12} - \frac{1}{2} a^{11} - \frac{11}{24} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{24} a^{20} + \frac{1}{24} a^{17} - \frac{1}{8} a^{16} - \frac{1}{4} a^{15} - \frac{5}{24} a^{14} - \frac{1}{8} a^{13} + \frac{7}{24} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{21} - \frac{1}{48} a^{19} + \frac{1}{48} a^{18} - \frac{1}{16} a^{17} - \frac{1}{48} a^{16} - \frac{1}{6} a^{15} + \frac{3}{16} a^{14} - \frac{7}{48} a^{13} + \frac{1}{12} a^{12} + \frac{5}{16} a^{11} + \frac{23}{48} a^{10} + \frac{1}{16} a^{9} - \frac{3}{16} a^{8} + \frac{1}{8} a^{7} - \frac{7}{16} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{6} a^{3} + \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{48} a^{22} - \frac{1}{48} a^{20} - \frac{1}{48} a^{19} + \frac{1}{48} a^{18} - \frac{1}{48} a^{17} + \frac{1}{24} a^{16} + \frac{7}{48} a^{15} + \frac{5}{48} a^{14} - \frac{5}{24} a^{13} - \frac{11}{48} a^{12} + \frac{23}{48} a^{11} - \frac{11}{48} a^{10} - \frac{11}{48} a^{9} + \frac{3}{8} a^{8} + \frac{5}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{12} a^{4} + \frac{1}{12} a^{2} + \frac{5}{12} a + \frac{1}{6}$, $\frac{1}{1827309298827628347823445856} a^{23} - \frac{1473817888801727544648093}{152275774902302362318620488} a^{22} + \frac{561732763673785791050725}{228413662353453543477930732} a^{21} + \frac{18766426272289680202899}{609103099609209449274481952} a^{20} + \frac{1711180052499363449541419}{456827324706907086955861464} a^{19} + \frac{1354392773755357364927629}{152275774902302362318620488} a^{18} - \frac{66089784628356298472352377}{609103099609209449274481952} a^{17} - \frac{902392813745836878754898}{57103415588363385869482683} a^{16} - \frac{29400971099169878176702333}{609103099609209449274481952} a^{15} + \frac{131466160437149537603220479}{609103099609209449274481952} a^{14} - \frac{6826950245498569514197235}{57103415588363385869482683} a^{13} + \frac{71099033323294796939377881}{609103099609209449274481952} a^{12} + \frac{309828025821753902644921297}{913654649413814173911722928} a^{11} + \frac{205668700153897323799937713}{913654649413814173911722928} a^{10} - \frac{98350626335332943114929607}{1827309298827628347823445856} a^{9} + \frac{138314270426675456649363415}{304551549804604724637240976} a^{8} - \frac{27271659641953150581831749}{152275774902302362318620488} a^{7} + \frac{189062124276697438432606919}{609103099609209449274481952} a^{6} - \frac{360345345202777521763306397}{913654649413814173911722928} a^{5} - \frac{86552670562055707660426831}{304551549804604724637240976} a^{4} - \frac{27531235076728595814907139}{456827324706907086955861464} a^{3} + \frac{120072835544684775292226263}{456827324706907086955861464} a^{2} + \frac{127966509442727737961843197}{456827324706907086955861464} a + \frac{222638534728156948463669519}{456827324706907086955861464}$
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: | \( 494355842.3116425 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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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
6.2.170772624.2, 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: | 24.4.7654476884289954721505578121232384.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | $20{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{8}$ | $20{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ | ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ | $20{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
$2$ | 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.6.4.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
$3$ | 3.12.12.5 | $x^{12} + 33 x^{11} - 63 x^{10} - 36 x^{9} - 90 x^{8} - 54 x^{7} - 54 x^{6} - 108 x^{4} - 27 x^{3} - 81 x^{2} + 81 x - 81$ | $3$ | $4$ | $12$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ |
3.12.14.9 | $x^{12} + 6 x^{11} + 12 x^{10} - 3 x^{9} + 12 x^{6} - 9 x^{5} + 9 x^{4} - 9 x^{3} - 9 x^{2} - 9$ | $6$ | $2$ | $14$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ | |
11 | Data not computed |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.13068.120.a.a | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 24.4.7654476884289954721505578121232384.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
2.13068.120.a.b | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 24.4.7654476884289954721505578121232384.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
2.13068.120.a.c | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 24.4.7654476884289954721505578121232384.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
2.13068.120.a.d | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 24.4.7654476884289954721505578121232384.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
* | 3.13068.12t76.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 10.0.67507613675568.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
* | 3.13068.12t76.a.b | $3$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 10.0.67507613675568.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
3.39204.12t33.a.a | $3$ | $ 2^{2} \cdot 3^{4} \cdot 11^{2}$ | 5.1.4743684.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
3.39204.12t33.a.b | $3$ | $ 2^{2} \cdot 3^{4} \cdot 11^{2}$ | 5.1.4743684.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
4.4743684.10t11.a.a | $4$ | $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ | 10.0.67507613675568.1 | $A_5\times C_2$ (as 10T11) | $1$ | $0$ | |
4.4743684.5t4.a.a | $4$ | $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ | 5.1.4743684.1 | $A_5$ (as 5T4) | $1$ | $0$ | |
4.4743684.40t188.a.a | $4$ | $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ | 24.4.7654476884289954721505578121232384.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
4.4743684.40t188.a.b | $4$ | $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ | 24.4.7654476884289954721505578121232384.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ | |
5.512317872.12t75.a.a | $5$ | $ 2^{4} \cdot 3^{7} \cdot 11^{4}$ | 10.0.67507613675568.1 | $A_5\times C_2$ (as 10T11) | $1$ | $-1$ | |
* | 5.170772624.6t12.a.a | $5$ | $ 2^{4} \cdot 3^{6} \cdot 11^{4}$ | 5.1.4743684.1 | $A_5$ (as 5T4) | $1$ | $1$ |
* | 6.512317872.24t576.a.a | $6$ | $ 2^{4} \cdot 3^{7} \cdot 11^{4}$ | 24.4.7654476884289954721505578121232384.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ |
* | 6.512317872.24t576.a.b | $6$ | $ 2^{4} \cdot 3^{7} \cdot 11^{4}$ | 24.4.7654476884289954721505578121232384.2 | $C_4.A_5$ (as 24T576) | $0$ | $0$ |