Normalized defining polynomial
\( x^{21} - 48 x^{19} - 32 x^{18} + 846 x^{17} + 1128 x^{16} - 5726 x^{15} - 12204 x^{14} - 3924 x^{13} + 9424 x^{12} + 186678 x^{11} + 589812 x^{10} - 71609 x^{9} - 2888964 x^{8} - 4329267 x^{7} + 474058 x^{6} + 8477028 x^{5} + 11551176 x^{4} + 8074864 x^{3} + 3259872 x^{2} + 724416 x + 68992 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[17, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(10322551559300386187871072331953642204421617893376=2^{14}\cdot 3^{22}\cdot 7^{4}\cdot 11^{2}\cdot 1601^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $215.77$ | 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: | $2, 3, 7, 11, 1601$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{16} + \frac{1}{32} a^{15} - \frac{5}{16} a^{14} - \frac{1}{8} a^{13} - \frac{9}{32} a^{12} + \frac{1}{16} a^{11} - \frac{3}{32} a^{10} + \frac{1}{8} a^{9} + \frac{3}{16} a^{8} + \frac{1}{8} a^{7} + \frac{3}{32} a^{6} + \frac{23}{64} a^{4} + \frac{5}{32} a^{3} - \frac{23}{64} a^{2} - \frac{1}{16} a - \frac{7}{16}$, $\frac{1}{512} a^{17} - \frac{3}{64} a^{15} - \frac{3}{16} a^{14} - \frac{97}{256} a^{13} - \frac{19}{64} a^{12} + \frac{121}{256} a^{11} + \frac{37}{128} a^{10} + \frac{63}{128} a^{9} - \frac{5}{32} a^{8} - \frac{101}{256} a^{7} - \frac{51}{128} a^{6} + \frac{151}{512} a^{5} - \frac{41}{128} a^{4} + \frac{21}{512} a^{3} + \frac{53}{256} a^{2} + \frac{43}{128} a - \frac{9}{64}$, $\frac{1}{315392} a^{18} - \frac{1}{2048} a^{17} + \frac{85}{39424} a^{16} + \frac{1209}{19712} a^{15} - \frac{1537}{157696} a^{14} + \frac{31495}{78848} a^{13} - \frac{2617}{22528} a^{12} + \frac{291}{616} a^{11} + \frac{17373}{78848} a^{10} - \frac{2301}{39424} a^{9} - \frac{34005}{157696} a^{8} - \frac{1513}{39424} a^{7} + \frac{133071}{315392} a^{6} + \frac{24643}{157696} a^{5} + \frac{67069}{315392} a^{4} + \frac{13801}{39424} a^{3} - \frac{119}{512} a^{2} + \frac{9}{64} a - \frac{7}{256}$, $\frac{1}{2523136} a^{19} - \frac{1}{630784} a^{18} + \frac{555}{630784} a^{17} + \frac{3}{2464} a^{16} - \frac{68561}{1261568} a^{15} - \frac{60369}{157696} a^{14} + \frac{101477}{1261568} a^{13} - \frac{6117}{630784} a^{12} - \frac{119299}{630784} a^{11} + \frac{15857}{157696} a^{10} - \frac{202317}{1261568} a^{9} + \frac{304839}{630784} a^{8} - \frac{667361}{2523136} a^{7} + \frac{64205}{157696} a^{6} - \frac{44445}{229376} a^{5} - \frac{29885}{1261568} a^{4} - \frac{32797}{315392} a^{3} - \frac{721}{2048} a^{2} + \frac{513}{2048} a + \frac{99}{1024}$, $\frac{1}{20185088} a^{20} + \frac{1}{10092544} a^{19} + \frac{5}{5046272} a^{18} - \frac{415}{2523136} a^{17} - \frac{10705}{10092544} a^{16} + \frac{21039}{720896} a^{15} - \frac{3236427}{10092544} a^{14} + \frac{664421}{2523136} a^{13} + \frac{299615}{5046272} a^{12} + \frac{42399}{360448} a^{11} - \frac{9147}{1441792} a^{10} - \frac{20521}{157696} a^{9} + \frac{7665863}{20185088} a^{8} - \frac{1684027}{10092544} a^{7} + \frac{9899937}{20185088} a^{6} + \frac{677571}{5046272} a^{5} - \frac{410129}{5046272} a^{4} + \frac{369}{28672} a^{3} + \frac{6299}{16384} a^{2} + \frac{339}{4096} a - \frac{1719}{4096}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $18$ | 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: | \( 523529356224000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1959552 |
| The 171 conjugacy class representatives for t21n122 are not computed |
| Character table for t21n122 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 | $21$ | R | R | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$ |
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 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.34 | $x^{14} - x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{7} + 2 x^{4} + 2 x^{2} + 2 x + 1$ | $2$ | $7$ | $14$ | 14T21 | $[2, 2, 2, 2, 2, 2]^{7}$ | |
| $3$ | 3.3.4.3 | $x^{3} - 3 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.6.6.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 18$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
| 3.6.6.5 | $x^{6} + 6 x^{3} + 9 x^{2} + 9$ | $3$ | $2$ | $6$ | $S_3^2$ | $[3/2, 3/2]_{2}^{2}$ | |
| 3.6.6.1 | $x^{6} + 3 x^{5} - 2$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.6.4.1 | $x^{6} + 35 x^{3} + 441$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.3.2.1 | $x^{3} - 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.6.0.1 | $x^{6} + x^{2} - 2 x + 8$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 1601 | Data not computed | ||||||