Normalized defining polynomial
\( x^{21} - 6 x^{19} - 4 x^{18} - 126 x^{17} - 168 x^{16} + 862 x^{15} + 1836 x^{14} + 1548 x^{13} + 1136 x^{12} - 8370 x^{11} - 30396 x^{10} - 35873 x^{9} - 6948 x^{8} + 27087 x^{7} + 39230 x^{6} + 35532 x^{5} + 26712 x^{4} + 15568 x^{3} + 6048 x^{2} + 1344 x + 128 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[13, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(9211785278619203955690014672704286619549696=2^{14}\cdot 3^{19}\cdot 7\cdot 1601^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $111.15$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 7, 1601$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{16} + \frac{1}{32} a^{15} + \frac{11}{32} a^{14} - \frac{3}{8} a^{13} - \frac{7}{32} a^{12} + \frac{7}{16} a^{11} - \frac{13}{32} a^{10} - \frac{1}{8} a^{9} - \frac{5}{16} a^{8} - \frac{3}{8} a^{7} + \frac{15}{32} a^{6} - \frac{17}{64} a^{4} + \frac{13}{32} a^{3} + \frac{11}{64} a^{2} - \frac{3}{16} a - \frac{5}{16}$, $\frac{1}{512} a^{17} + \frac{9}{256} a^{15} - \frac{17}{128} a^{14} - \frac{79}{256} a^{13} + \frac{15}{64} a^{12} + \frac{119}{256} a^{11} - \frac{53}{128} a^{10} + \frac{47}{128} a^{9} - \frac{7}{32} a^{8} + \frac{71}{256} a^{7} + \frac{1}{128} a^{6} + \frac{111}{512} a^{5} + \frac{47}{128} a^{4} - \frac{41}{512} a^{3} - \frac{17}{256} a^{2} + \frac{49}{128} a + \frac{21}{64}$, $\frac{1}{315392} a^{18} - \frac{1}{2048} a^{17} - \frac{951}{157696} a^{16} + \frac{573}{39424} a^{15} - \frac{15707}{157696} a^{14} - \frac{2335}{78848} a^{13} + \frac{4713}{22528} a^{12} - \frac{39}{308} a^{11} + \frac{26065}{78848} a^{10} + \frac{7631}{39424} a^{9} - \frac{65033}{157696} a^{8} + \frac{10323}{39424} a^{7} + \frac{72967}{315392} a^{6} - \frac{44805}{157696} a^{5} + \frac{66623}{315392} a^{4} - \frac{11855}{39424} a^{3} + \frac{18391}{39424} a^{2} - \frac{1525}{4928} a - \frac{1713}{19712}$, $\frac{1}{2523136} a^{19} - \frac{1}{630784} a^{18} - \frac{181}{1261568} a^{17} - \frac{1187}{630784} a^{16} + \frac{37341}{1261568} a^{15} + \frac{29247}{78848} a^{14} - \frac{564021}{1261568} a^{13} + \frac{24223}{57344} a^{12} + \frac{33969}{630784} a^{11} - \frac{4317}{22528} a^{10} + \frac{364191}{1261568} a^{9} - \frac{145661}{630784} a^{8} - \frac{108047}{360448} a^{7} - \frac{367}{157696} a^{6} + \frac{398883}{2523136} a^{5} + \frac{188857}{1261568} a^{4} - \frac{43067}{315392} a^{3} + \frac{1783}{22528} a^{2} - \frac{7583}{22528} a + \frac{5813}{78848}$, $\frac{1}{60555264} a^{20} + \frac{5}{30277632} a^{19} + \frac{47}{30277632} a^{18} - \frac{409}{2523136} a^{17} - \frac{55805}{10092544} a^{16} + \frac{163585}{5046272} a^{15} + \frac{13557227}{30277632} a^{14} - \frac{2418895}{7569408} a^{13} + \frac{6059863}{15138816} a^{12} + \frac{94187}{2523136} a^{11} + \frac{268101}{10092544} a^{10} - \frac{153827}{1261568} a^{9} + \frac{3971647}{60555264} a^{8} + \frac{7565225}{30277632} a^{7} + \frac{24113923}{60555264} a^{6} + \frac{2028573}{5046272} a^{5} + \frac{2334643}{5046272} a^{4} + \frac{54419}{315392} a^{3} + \frac{1311157}{3784704} a^{2} - \frac{29815}{86016} a - \frac{247565}{946176}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $16$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 388660756503000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 3919104 |
| The 288 conjugacy class representatives for t21n131 are not computed |
| Character table for t21n131 is not computed |
Intermediate fields
| 7.7.4103684801.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 42 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 | ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }$ | $21$ | $21$ | ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.7.0.1 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.16 | $x^{14} - 2 x^{13} - 3 x^{12} - 2 x^{10} + 4 x^{9} + 2 x^{8} - 2 x^{7} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 2 x + 3$ | $2$ | $7$ | $14$ | $C_2 \wr C_7$ | $[2, 2, 2, 2, 2, 2]^{14}$ | |
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.6.6.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 18$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
| 3.6.6.1 | $x^{6} + 3 x^{5} - 2$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
| 3.6.6.5 | $x^{6} + 6 x^{3} + 9 x^{2} + 9$ | $3$ | $2$ | $6$ | $S_3^2$ | $[3/2, 3/2]_{2}^{2}$ | |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 1601 | Data not computed | ||||||