Normalized defining polynomial
\( x^{24} - 6 x^{23} + 23 x^{22} - 58 x^{21} + 100 x^{20} - 160 x^{19} + 242 x^{18} - 156 x^{17} - 228 x^{16} + 452 x^{15} + 164 x^{14} - 1190 x^{13} + 1288 x^{12} - 326 x^{11} - 800 x^{10} + 832 x^{9} + 514 x^{8} - 1302 x^{7} + 500 x^{6} + 664 x^{5} - 155 x^{4} - 1418 x^{3} + 1951 x^{2} - 1112 x + 241 \)
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: | \(18075490334784000000000000000000\)\(\medspace = 2^{24}\cdot 3^{24}\cdot 5^{18}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $20.06$ | 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: | $2, 3, 5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{19} a^{20} - \frac{4}{19} a^{19} - \frac{7}{19} a^{18} - \frac{7}{19} a^{17} + \frac{7}{19} a^{16} + \frac{6}{19} a^{15} + \frac{9}{19} a^{14} + \frac{1}{19} a^{13} - \frac{6}{19} a^{12} + \frac{9}{19} a^{11} + \frac{7}{19} a^{10} - \frac{3}{19} a^{9} + \frac{2}{19} a^{8} + \frac{2}{19} a^{7} - \frac{6}{19} a^{6} - \frac{5}{19} a^{5} - \frac{3}{19} a^{4} + \frac{9}{19} a^{2} + \frac{1}{19} a - \frac{3}{19}$, $\frac{1}{19} a^{21} - \frac{4}{19} a^{19} + \frac{3}{19} a^{18} - \frac{2}{19} a^{17} - \frac{4}{19} a^{16} - \frac{5}{19} a^{15} - \frac{1}{19} a^{14} - \frac{2}{19} a^{13} + \frac{4}{19} a^{12} + \frac{5}{19} a^{11} + \frac{6}{19} a^{10} + \frac{9}{19} a^{9} - \frac{9}{19} a^{8} + \frac{2}{19} a^{7} + \frac{9}{19} a^{6} - \frac{4}{19} a^{5} + \frac{7}{19} a^{4} + \frac{9}{19} a^{3} - \frac{1}{19} a^{2} + \frac{1}{19} a + \frac{7}{19}$, $\frac{1}{520391} a^{22} + \frac{4421}{520391} a^{21} - \frac{652}{520391} a^{20} - \frac{229390}{520391} a^{19} + \frac{251383}{520391} a^{18} - \frac{42842}{520391} a^{17} + \frac{253826}{520391} a^{16} + \frac{244452}{520391} a^{15} - \frac{93874}{520391} a^{14} + \frac{241295}{520391} a^{13} - \frac{55259}{520391} a^{12} - \frac{117101}{520391} a^{11} - \frac{253938}{520391} a^{10} + \frac{4691}{27389} a^{9} - \frac{143170}{520391} a^{8} + \frac{79622}{520391} a^{7} + \frac{228524}{520391} a^{6} - \frac{3747}{8531} a^{5} + \frac{228049}{520391} a^{4} + \frac{5531}{520391} a^{3} - \frac{106620}{520391} a^{2} - \frac{195112}{520391} a - \frac{231627}{520391}$, $\frac{1}{419956795432079825530364746485222889} a^{23} - \frac{101319215942595358359036661784}{419956795432079825530364746485222889} a^{22} - \frac{10748810243223182880580531160791426}{419956795432079825530364746485222889} a^{21} + \frac{8573334410105795729091173607438655}{419956795432079825530364746485222889} a^{20} + \frac{2835713908526824366363550393916869}{22102989233267359238440249815011731} a^{19} - \frac{186833622028223032850129027066324130}{419956795432079825530364746485222889} a^{18} + \frac{103163242683464533647967380650929371}{419956795432079825530364746485222889} a^{17} + \frac{25003593580737856436994456804448162}{419956795432079825530364746485222889} a^{16} + \frac{23820997075217078695098363725759847}{419956795432079825530364746485222889} a^{15} + \frac{9897372513393668559421471034012683}{419956795432079825530364746485222889} a^{14} - \frac{16664015453315454594600495449082397}{419956795432079825530364746485222889} a^{13} - \frac{32223800907723303119921687129274460}{419956795432079825530364746485222889} a^{12} + \frac{146723562476124758328321705184496754}{419956795432079825530364746485222889} a^{11} - \frac{569496638052847335137654019400330}{14481268808002752604495336085697341} a^{10} - \frac{85508343340696381121036966465498166}{419956795432079825530364746485222889} a^{9} - \frac{154563330902656688521538083163387636}{419956795432079825530364746485222889} a^{8} + \frac{30953999633935022841384750801846882}{419956795432079825530364746485222889} a^{7} - \frac{140061901963098996446282679986627948}{419956795432079825530364746485222889} a^{6} - \frac{21131501790739865450536250494917081}{419956795432079825530364746485222889} a^{5} - \frac{172380349097524661189657697362642735}{419956795432079825530364746485222889} a^{4} + \frac{106564340816968229368280771930135870}{419956795432079825530364746485222889} a^{3} + \frac{109633136719095248228510667763892122}{419956795432079825530364746485222889} a^{2} - \frac{113124710509798825189788672028592445}{419956795432079825530364746485222889} a + \frac{22916780383218740968384709922429}{91713648270818918001826762717891}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( \frac{73616946842978946}{385899254913200159} a^{23} - \frac{7462656356153571980}{7332085843350803021} a^{22} + \frac{27199219556983244939}{7332085843350803021} a^{21} - \frac{63003113424945170569}{7332085843350803021} a^{20} + \frac{97853678341237425469}{7332085843350803021} a^{19} - \frac{158509176010303132348}{7332085843350803021} a^{18} + \frac{232899148742777208807}{7332085843350803021} a^{17} - \frac{62853260300608451637}{7332085843350803021} a^{16} - \frac{361334797096169948987}{7332085843350803021} a^{15} + \frac{390773395557468131143}{7332085843350803021} a^{14} + \frac{491465161484057999336}{7332085843350803021} a^{13} - \frac{1336747531522870070326}{7332085843350803021} a^{12} + \frac{907671804585560377208}{7332085843350803021} a^{11} + \frac{129018987937036181}{6326217293659019} a^{10} - \frac{1014351005174727814448}{7332085843350803021} a^{9} + \frac{485370744649705117064}{7332085843350803021} a^{8} + \frac{1045162766334852803796}{7332085843350803021} a^{7} - \frac{1122011553221285078722}{7332085843350803021} a^{6} - \frac{55062761327452545061}{7332085843350803021} a^{5} + \frac{886012307751874158164}{7332085843350803021} a^{4} + \frac{19877686242970701284}{385899254913200159} a^{3} - \frac{1734124202687164364017}{7332085843350803021} a^{2} + \frac{82737670931200852412}{385899254913200159} a - \frac{508948454126450352214}{7332085843350803021} \) (order $4$) | 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: | \( 535957.6069656424 \) (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
| \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{15})^+\), 4.0.18000.1, \(\Q(i, \sqrt{5})\), 6.0.648000.1, 8.0.324000000.1, 12.0.419904000000.1 |
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 | R | R | R | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.12.18.61 | $x^{12} - 12 x^{10} - 27 x^{9} + 18 x^{8} - 36 x^{7} - 33 x^{6} - 27 x^{5} + 27 x^{4} + 27 x^{3} + 27 x + 18$ | $6$ | $2$ | $18$ | $C_{12}$ | $[2]_{2}^{2}$ |
| 3.12.6.1 | $x^{12} - 243 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ | |
| $5$ | 5.12.9.1 | $x^{12} - 10 x^{8} - 375 x^{4} - 2000$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 5.12.9.1 | $x^{12} - 10 x^{8} - 375 x^{4} - 2000$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.9.3t1.a.a | $1$ | $ 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) | $0$ | $1$ | |
| 1.180.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.52488000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.180.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.52488000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.36.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.36.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.60.4t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | 4.0.18000.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.60.4t1.a.b | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | 4.0.18000.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.180.12t1.a.a | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.3099363912000000000.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.b.a | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.b.b | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.180.12t1.a.b | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.3099363912000000000.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.180.12t1.a.c | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.3099363912000000000.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.b.c | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.180.12t1.a.d | $1$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.3099363912000000000.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.b.d | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 2.1620.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5 $ | 3.1.1620.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.1620.6t3.g.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5 $ | 6.2.13122000.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.180.12t18.a.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.419904000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.180.12t18.a.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 12.0.419904000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.180.6t5.b.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.648000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.180.6t5.b.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 5 $ | 6.0.648000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| 2.8100.12t11.a.a | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ | 12.4.193710244500000000.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.8100.12t11.a.b | $2$ | $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ | 12.4.193710244500000000.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.2700.24t65.a.a | $2$ | $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ | 24.0.18075490334784000000000000000000.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.2700.24t65.a.b | $2$ | $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ | 24.0.18075490334784000000000000000000.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.2700.24t65.a.c | $2$ | $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ | 24.0.18075490334784000000000000000000.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.2700.24t65.a.d | $2$ | $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ | 24.0.18075490334784000000000000000000.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |