Normalized defining polynomial
\(x^{24} - 8 x^{23} + 35 x^{22} - 105 x^{21} + 233 x^{20} - 377 x^{19} + 373 x^{18} + 86 x^{17} - 1310 x^{16} + 3226 x^{15} - 4893 x^{14} + 4424 x^{13} - 205 x^{12} - 6944 x^{11} + 13153 x^{10} - 14094 x^{9} + 9116 x^{8} - 2660 x^{7} - 525 x^{6} + 689 x^{5} - 175 x^{4} - 37 x^{3} + 37 x^{2} - 10 x + 1\) 
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: | \(1729054511370401309967041015625\)\(\medspace = 3^{12}\cdot 5^{18}\cdot 31^{8}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $18.19$ | 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: | $3, 5, 31$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{29} a^{20} - \frac{6}{29} a^{19} + \frac{14}{29} a^{18} - \frac{14}{29} a^{17} - \frac{3}{29} a^{16} + \frac{8}{29} a^{15} + \frac{14}{29} a^{14} + \frac{1}{29} a^{13} - \frac{2}{29} a^{12} + \frac{12}{29} a^{11} - \frac{14}{29} a^{10} + \frac{8}{29} a^{9} + \frac{14}{29} a^{8} + \frac{3}{29} a^{7} + \frac{12}{29} a^{6} - \frac{14}{29} a^{5} + \frac{2}{29} a^{4} - \frac{12}{29} a^{3} - \frac{14}{29} a^{2} - \frac{10}{29} a - \frac{1}{29}$, $\frac{1}{899} a^{21} + \frac{5}{899} a^{20} + \frac{209}{899} a^{19} + \frac{314}{899} a^{18} - \frac{418}{899} a^{17} + \frac{381}{899} a^{16} + \frac{131}{899} a^{15} + \frac{5}{29} a^{14} + \frac{154}{899} a^{13} - \frac{10}{899} a^{12} - \frac{172}{899} a^{11} + \frac{376}{899} a^{10} - \frac{391}{899} a^{9} - \frac{336}{899} a^{8} + \frac{364}{899} a^{7} + \frac{2}{899} a^{6} - \frac{210}{899} a^{5} - \frac{222}{899} a^{4} - \frac{88}{899} a^{3} - \frac{193}{899} a^{2} - \frac{169}{899} a - \frac{69}{899}$, $\frac{1}{899} a^{22} - \frac{2}{899} a^{20} + \frac{385}{899} a^{19} - \frac{97}{899} a^{18} - \frac{11}{31} a^{17} - \frac{317}{899} a^{16} - \frac{190}{899} a^{15} + \frac{371}{899} a^{14} - \frac{67}{899} a^{13} + \frac{250}{899} a^{12} - \frac{97}{899} a^{11} + \frac{333}{899} a^{10} + \frac{131}{899} a^{9} + \frac{339}{899} a^{8} + \frac{321}{899} a^{7} + \frac{245}{899} a^{6} - \frac{164}{899} a^{5} - \frac{249}{899} a^{4} - \frac{218}{899} a^{3} - \frac{196}{899} a^{2} - \frac{61}{899} a - \frac{368}{899}$, $\frac{1}{11868543608143540911421} a^{23} + \frac{4684544303211865073}{11868543608143540911421} a^{22} - \frac{2257035256652470824}{11868543608143540911421} a^{21} - \frac{1427868117259036515}{409260124418742790049} a^{20} + \frac{154046087434159728435}{409260124418742790049} a^{19} + \frac{2497372221107995246562}{11868543608143540911421} a^{18} - \frac{3090431714232401800518}{11868543608143540911421} a^{17} + \frac{2429954813607995706858}{11868543608143540911421} a^{16} + \frac{587452496068687244334}{11868543608143540911421} a^{15} - \frac{515098462574121588234}{11868543608143540911421} a^{14} - \frac{4885307908756681871693}{11868543608143540911421} a^{13} + \frac{794653194182218182895}{11868543608143540911421} a^{12} - \frac{2316614783254686125593}{11868543608143540911421} a^{11} + \frac{470858823030795760400}{11868543608143540911421} a^{10} + \frac{1081886529545810688287}{11868543608143540911421} a^{9} - \frac{125200598313107149079}{382856245423985190691} a^{8} + \frac{5607780455201748749742}{11868543608143540911421} a^{7} - \frac{5702291777162463598143}{11868543608143540911421} a^{6} + \frac{5776408298278631542732}{11868543608143540911421} a^{5} - \frac{2452895507991068366958}{11868543608143540911421} a^{4} + \frac{341043506318449358776}{11868543608143540911421} a^{3} + \frac{4924220837334206800113}{11868543608143540911421} a^{2} + \frac{513603823395221872971}{11868543608143540911421} a - \frac{4249438757793593569172}{11868543608143540911421}$
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: | \( -\frac{7734100806656053550843}{409260124418742790049} a^{23} + \frac{1693683224712430097540188}{11868543608143540911421} a^{22} - \frac{7090008489812080887569001}{11868543608143540911421} a^{21} + \frac{20368041693402810603857679}{11868543608143540911421} a^{20} - \frac{43116509401361366582348538}{11868543608143540911421} a^{19} + \frac{65201825978955000247187270}{11868543608143540911421} a^{18} - \frac{54389259790109903136846857}{11868543608143540911421} a^{17} - \frac{43706410432866229062553548}{11868543608143540911421} a^{16} + \frac{274199302232129549703642975}{11868543608143540911421} a^{15} - \frac{600461831108203963506430442}{11868543608143540911421} a^{14} + \frac{827882160417212094168984537}{11868543608143540911421} a^{13} - \frac{620600631723841736157137884}{11868543608143540911421} a^{12} - \frac{232597302108791818065380974}{11868543608143540911421} a^{11} + \frac{1452958002905821036039530273}{11868543608143540911421} a^{10} - \frac{2297679537017594890047330281}{11868543608143540911421} a^{9} + \frac{2368871237152426530958445}{13201939497378799679} a^{8} - \frac{1088778805324366041663867783}{11868543608143540911421} a^{7} + \frac{3731875305378225621736111}{409260124418742790049} a^{6} + \frac{165980947158962287342034348}{11868543608143540911421} a^{5} - \frac{80030226322022007352319547}{11868543608143540911421} a^{4} + \frac{3587790200741572434146851}{11868543608143540911421} a^{3} + \frac{9838687081568899957579250}{11868543608143540911421} a^{2} - \frac{3879677771549136926752282}{11868543608143540911421} a + \frac{499762829749894184209711}{11868543608143540911421} \) (order $30$)
| 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: | \( 2176553.0653759805 \) (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
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.3243375.1, \(\Q(\zeta_{15})\), 12.0.10519481390625.2 |
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 | ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ | R | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $31$ | $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.3.2.1 | $x^{3} - 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.2.1 | $x^{3} - 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.2.1 | $x^{3} - 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.2.1 | $x^{3} - 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
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) | $1$ | $-1$ |
| * | 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.93.6t1.a.a | $1$ | $ 3 \cdot 31 $ | 6.0.24935067.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.31.3t1.a.a | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.93.6t1.a.b | $1$ | $ 3 \cdot 31 $ | 6.0.24935067.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.155.6t1.a.a | $1$ | $ 5 \cdot 31 $ | 6.6.115440125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.465.6t1.b.a | $1$ | $ 3 \cdot 5 \cdot 31 $ | 6.0.3116883375.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.155.6t1.a.b | $1$ | $ 5 \cdot 31 $ | 6.6.115440125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.465.6t1.b.b | $1$ | $ 3 \cdot 5 \cdot 31 $ | 6.0.3116883375.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.31.3t1.a.b | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| * | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| 1.465.12t1.a.a | $1$ | $ 3 \cdot 5 \cdot 31 $ | 12.12.1214370246668923828125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.465.12t1.a.b | $1$ | $ 3 \cdot 5 \cdot 31 $ | 12.12.1214370246668923828125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.465.12t1.a.c | $1$ | $ 3 \cdot 5 \cdot 31 $ | 12.12.1214370246668923828125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.465.12t1.a.d | $1$ | $ 3 \cdot 5 \cdot 31 $ | 12.12.1214370246668923828125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.155.12t1.a.a | $1$ | $ 5 \cdot 31 $ | 12.0.1665802807501953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.155.12t1.a.b | $1$ | $ 5 \cdot 31 $ | 12.0.1665802807501953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.155.12t1.a.c | $1$ | $ 5 \cdot 31 $ | 12.0.1665802807501953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.155.12t1.a.d | $1$ | $ 5 \cdot 31 $ | 12.0.1665802807501953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.14415.3t2.a.a | $2$ | $ 3 \cdot 5 \cdot 31^{2}$ | 3.1.14415.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.14415.6t3.e.a | $2$ | $ 3 \cdot 5 \cdot 31^{2}$ | 6.0.623376675.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.465.6t5.c.a | $2$ | $ 3 \cdot 5 \cdot 31 $ | 6.0.3243375.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.465.6t5.c.b | $2$ | $ 3 \cdot 5 \cdot 31 $ | 6.0.3243375.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.465.12t18.a.a | $2$ | $ 3 \cdot 5 \cdot 31 $ | 12.0.10519481390625.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.465.12t18.a.b | $2$ | $ 3 \cdot 5 \cdot 31 $ | 12.0.10519481390625.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| 2.72075.12t11.b.a | $2$ | $ 3 \cdot 5^{2} \cdot 31^{2}$ | 12.0.134930027407658203125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.72075.12t11.b.b | $2$ | $ 3 \cdot 5^{2} \cdot 31^{2}$ | 12.0.134930027407658203125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.2325.24t65.a.a | $2$ | $ 3 \cdot 5^{2} \cdot 31 $ | 24.0.1729054511370401309967041015625.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.2325.24t65.a.b | $2$ | $ 3 \cdot 5^{2} \cdot 31 $ | 24.0.1729054511370401309967041015625.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.2325.24t65.a.c | $2$ | $ 3 \cdot 5^{2} \cdot 31 $ | 24.0.1729054511370401309967041015625.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.2325.24t65.a.d | $2$ | $ 3 \cdot 5^{2} \cdot 31 $ | 24.0.1729054511370401309967041015625.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |