Normalized defining polynomial
\(x^{24} - 5 x^{23} + 7 x^{22} + 9 x^{21} - 43 x^{20} + 66 x^{19} - 38 x^{18} - 93 x^{17} + 491 x^{16} - 1291 x^{15} + 2386 x^{14} - 3246 x^{13} + 3278 x^{12} - 2354 x^{11} + 1061 x^{10} - 234 x^{9} + 126 x^{8} - 282 x^{7} + 277 x^{6} - 136 x^{5} + 42 x^{4} - 19 x^{3} + 12 x^{2} - 5 x + 1\) 
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(2272880662621757205963134765625\)\(\medspace = 3^{16}\cdot 5^{18}\cdot 7^{12}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $18.40$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $3, 5, 7$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $12$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{33} a^{20} - \frac{10}{33} a^{19} - \frac{1}{33} a^{18} + \frac{1}{11} a^{17} - \frac{1}{33} a^{16} - \frac{3}{11} a^{13} + \frac{4}{11} a^{12} - \frac{4}{33} a^{11} - \frac{5}{33} a^{9} + \frac{3}{11} a^{8} - \frac{14}{33} a^{7} - \frac{2}{33} a^{5} - \frac{8}{33} a^{4} - \frac{4}{33} a^{3} - \frac{4}{33} a^{2} + \frac{1}{33} a - \frac{8}{33}$, $\frac{1}{33} a^{21} - \frac{2}{33} a^{19} - \frac{7}{33} a^{18} - \frac{4}{33} a^{17} - \frac{10}{33} a^{16} - \frac{3}{11} a^{14} - \frac{4}{11} a^{13} - \frac{16}{33} a^{12} - \frac{7}{33} a^{11} - \frac{5}{33} a^{10} - \frac{8}{33} a^{9} + \frac{10}{33} a^{8} - \frac{8}{33} a^{7} - \frac{2}{33} a^{6} + \frac{5}{33} a^{5} + \frac{5}{11} a^{4} - \frac{1}{3} a^{3} - \frac{2}{11} a^{2} + \frac{2}{33} a - \frac{14}{33}$, $\frac{1}{101013} a^{22} - \frac{252}{33671} a^{21} - \frac{928}{101013} a^{20} + \frac{45646}{101013} a^{19} - \frac{17117}{101013} a^{18} + \frac{39869}{101013} a^{17} + \frac{1853}{101013} a^{16} + \frac{3682}{33671} a^{15} + \frac{13869}{33671} a^{14} + \frac{527}{101013} a^{13} + \frac{44834}{101013} a^{12} + \frac{11148}{33671} a^{11} - \frac{44540}{101013} a^{10} + \frac{22370}{101013} a^{9} - \frac{29960}{101013} a^{8} + \frac{9902}{101013} a^{7} + \frac{20591}{101013} a^{6} - \frac{40193}{101013} a^{5} + \frac{25526}{101013} a^{4} + \frac{44981}{101013} a^{3} + \frac{40648}{101013} a^{2} - \frac{47860}{101013} a - \frac{36260}{101013}$, $\frac{1}{5263958949618330357} a^{23} + \frac{446888716732}{1754652983206110119} a^{22} - \frac{53471461417170409}{5263958949618330357} a^{21} + \frac{38908752377430407}{5263958949618330357} a^{20} - \frac{184771365167908300}{1754652983206110119} a^{19} + \frac{2543216934336938320}{5263958949618330357} a^{18} + \frac{655459795700444264}{5263958949618330357} a^{17} + \frac{1363593263966905535}{5263958949618330357} a^{16} + \frac{591957268213237827}{1754652983206110119} a^{15} - \frac{1560066643166320261}{5263958949618330357} a^{14} + \frac{406760728184272034}{5263958949618330357} a^{13} - \frac{28979148637427185}{1754652983206110119} a^{12} - \frac{57357385259033900}{1754652983206110119} a^{11} + \frac{2285119256973322268}{5263958949618330357} a^{10} + \frac{242911682527489040}{5263958949618330357} a^{9} - \frac{2006564879839306387}{5263958949618330357} a^{8} - \frac{330885627906896803}{1754652983206110119} a^{7} + \frac{11173687806447425}{478541722692575487} a^{6} - \frac{142937571897312063}{1754652983206110119} a^{5} + \frac{152900597827733354}{1754652983206110119} a^{4} - \frac{563907950687164742}{1754652983206110119} a^{3} + \frac{853990555216872748}{5263958949618330357} a^{2} - \frac{1978962535552469923}{5263958949618330357} a + \frac{1398125949338705584}{5263958949618330357}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -\frac{420542191288575801}{159513907564191829} a^{23} + \frac{1884319946847640178}{159513907564191829} a^{22} - \frac{1987664220605686358}{159513907564191829} a^{21} - \frac{4724789255142518635}{159513907564191829} a^{20} + \frac{1415299208988855822}{14501264324017439} a^{19} - \frac{19989949241358318038}{159513907564191829} a^{18} + \frac{6345062289037582560}{159513907564191829} a^{17} + \frac{41753912532463893142}{159513907564191829} a^{16} - \frac{185047803075614205948}{159513907564191829} a^{15} + \frac{449307406248197469246}{159513907564191829} a^{14} - \frac{779026316847090052525}{159513907564191829} a^{13} + \frac{980140157055206915327}{159513907564191829} a^{12} - \frac{899609150283943802129}{159513907564191829} a^{11} + \frac{554332968923387298268}{159513907564191829} a^{10} - \frac{179061393592224985443}{159513907564191829} a^{9} + \frac{9363005600615489148}{159513907564191829} a^{8} - \frac{41338870326134142235}{159513907564191829} a^{7} + \frac{92319816594770419231}{159513907564191829} a^{6} - \frac{70582448544126280591}{159513907564191829} a^{5} + \frac{24187758569682608476}{159513907564191829} a^{4} - \frac{5873819901447676869}{159513907564191829} a^{3} + \frac{4251476607857845654}{159513907564191829} a^{2} - \frac{2628319026083807795}{159513907564191829} a + \frac{812999923205803524}{159513907564191829} \) (order $10$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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)];
| |
| Regulator: | \( 1519657.3038774955 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_{12}\times S_3$ (as 24T65):
| A solvable group of order 72 |
| The 36 conjugacy class representatives for $C_{12}\times S_3$ |
| Character table for $C_{12}\times S_3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-35}) \), \(\Q(\zeta_{5})\), 4.4.6125.1, \(\Q(\sqrt{5}, \sqrt{-7})\), 6.0.3472875.1, 8.0.37515625.1, 12.0.12060860765625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 36 siblings: | data not computed |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | R | R | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.12.16.14 | $x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ | |
| 5 | Data not computed | ||||||
| 7 | Data not computed | ||||||
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.315.6t1.h.a | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.281302875.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.315.6t1.h.b | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.281302875.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.63.6t1.c.a | $1$ | $ 3^{2} \cdot 7 $ | 6.0.2250423.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.63.6t1.c.b | $1$ | $ 3^{2} \cdot 7 $ | 6.0.2250423.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.35.4t1.a.a | $1$ | $ 5 \cdot 7 $ | 4.4.6125.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.35.4t1.a.b | $1$ | $ 5 \cdot 7 $ | 4.4.6125.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.45.12t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.315.12t1.a.a | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.12.9891413435408203125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.315.12t1.a.b | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.12.9891413435408203125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.a.c | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.315.12t1.a.c | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.12.9891413435408203125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.315.12t1.a.d | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.12.9891413435408203125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.a.d | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.2835.3t2.a.a | $2$ | $ 3^{4} \cdot 5 \cdot 7 $ | 3.1.2835.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.2835.6t3.e.a | $2$ | $ 3^{4} \cdot 5 \cdot 7 $ | 6.0.56260575.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.315.12t18.a.a | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.0.12060860765625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.315.12t18.a.b | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.0.12060860765625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.315.6t5.a.a | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.3472875.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.315.6t5.a.b | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.3472875.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| 2.14175.12t11.a.a | $2$ | $ 3^{4} \cdot 5^{2} \cdot 7 $ | 12.0.201865580314453125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.14175.12t11.a.b | $2$ | $ 3^{4} \cdot 5^{2} \cdot 7 $ | 12.0.201865580314453125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.1575.24t65.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 24.0.2272880662621757205963134765625.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.1575.24t65.a.b | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 24.0.2272880662621757205963134765625.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.1575.24t65.a.c | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 24.0.2272880662621757205963134765625.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.1575.24t65.a.d | $2$ | $ 3^{2} \cdot 5^{2} \cdot 7 $ | 24.0.2272880662621757205963134765625.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |