Normalized defining polynomial
\( x^{21} - 7 x^{20} - 188 x^{19} + 778 x^{18} + 17508 x^{17} - 17843 x^{16} - 932140 x^{15} - 1542174 x^{14} + 26751880 x^{13} + 113453527 x^{12} - 285357185 x^{11} - 2903686327 x^{10} - 3836195996 x^{9} + 26214572726 x^{8} + 115945855524 x^{7} + 105703821028 x^{6} - 515904519138 x^{5} - 2086385055472 x^{4} - 3732486850641 x^{3} - 3805861147297 x^{2} - 2154688272800 x - 530346668697 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[5, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(602157249685783577798964485113044249656963472141=3^{10}\cdot 67^{2}\cdot 127^{2}\cdot 20731^{2}\cdot 121591^{2}\cdot 280909^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $188.46$ | 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: | $3, 67, 127, 20731, 121591, 280909$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{20} - \frac{20006897987340815051938402772521892240670881847370881136456822120023760570410}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{19} + \frac{12377306410359320403436471363963238713657139432593851630854749259013597165412}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{18} - \frac{55380078281449287841788261675489593477956065106611162867204177296488676952168}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{17} - \frac{32411498974181558807311192256526437999338122323188895586697286301792817503818}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{16} - \frac{54256719595485557766984715913247835781130238576063205604398685505007549869449}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{15} + \frac{1830264768669252317404556081995968895238231439568334984867340855184437811286}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{14} + \frac{41927683816964035983356831619196703578539835429093129179612994049639705521254}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{13} + \frac{31157983686300536593104992170693663828975678252331624516754833314568273666072}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{12} + \frac{50816008840310136196777939464944051098095223253542490183434177871558829870593}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{11} + \frac{15332800103553388637759702064292222415837879183399533472579720373035034158066}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{10} - \frac{54327365818197299835203697469894524577903302796447387602674402604455074938735}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{9} + \frac{21017151212970273488417730333945438765539469363132643302888549553867310081992}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{8} - \frac{31752567912752342604622615659905780433111109759385893350160009861638178488045}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{7} - \frac{58066492582753955447590974707639300937865206981499905385040449177441336499933}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{6} + \frac{49763230186336398305928455206657062747941453075390237531308369682672164758076}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{5} + \frac{28808047330115390686573220150853075737914150433763520120635984153585111998980}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{4} + \frac{17207850787567443770456549481051821404021694195072804530757478385825478324141}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{3} - \frac{37684900223218460577516160799883677793491275688314563180769008497768975133152}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a^{2} + \frac{9140016636386929675406163739474415631471751164548087789228489505075224373127}{120055575636198689658614925497127938390759557301808035673328289095778141336473} a + \frac{20147604823433898050791513163809323228281739928171292636335372317722614523091}{120055575636198689658614925497127938390759557301808035673328289095778141336473}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $12$ | 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: | \( 2553824201440000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 22044960 |
| The 261 conjugacy class representatives for t21n144 are not computed |
| Character table for t21n144 is not computed |
Intermediate fields
| 7.5.7584543.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | $15{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ | ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ | $15{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | $18{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | $18{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ | $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$ |
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 | ||||||
| $67$ | 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.3.2.3 | $x^{3} - 1072$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 67.6.0.1 | $x^{6} + x^{2} - x + 12$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 67.6.0.1 | $x^{6} + x^{2} - x + 12$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $127$ | 127.3.2.3 | $x^{3} - 10287$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 127.3.0.1 | $x^{3} - x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 127.6.0.1 | $x^{6} - x + 29$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 127.9.0.1 | $x^{9} - x + 26$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 20731 | Data not computed | ||||||
| 121591 | Data not computed | ||||||
| 280909 | Data not computed | ||||||