Normalized defining polynomial
\( x^{20} - x^{19} - x^{18} - 3 x^{17} - 117 x^{16} + 324 x^{15} + 544 x^{14} - 1944 x^{13} + 338 x^{12} + 1805 x^{11} - 3500 x^{10} + 3845 x^{9} - 784 x^{8} + 3698 x^{7} + 4586 x^{6} - 296 x^{5} - 8168 x^{4} - 9711 x^{3} + 4982 x^{2} + 2316 x - 1069 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(30596832887831976379669189453125=3^{4}\cdot 5^{15}\cdot 23^{4}\cdot 89^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.52$ | 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: | $3, 5, 23, 89$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{12791111916633188643745996987200262877133548129} a^{19} - \frac{3106577312906400216365820997906089342950811308}{12791111916633188643745996987200262877133548129} a^{18} - \frac{5257168067886512509376867869789289878194183173}{12791111916633188643745996987200262877133548129} a^{17} - \frac{174976714479122127168569930335572226781386135}{12791111916633188643745996987200262877133548129} a^{16} + \frac{5790115083104006455657434885478831735020866571}{12791111916633188643745996987200262877133548129} a^{15} - \frac{4951453868313865870002904259928803604487103710}{12791111916633188643745996987200262877133548129} a^{14} + \frac{3281967681173426853406798230300330815417430004}{12791111916633188643745996987200262877133548129} a^{13} - \frac{6202542483027406672452027600508408924169311830}{12791111916633188643745996987200262877133548129} a^{12} - \frac{3500109536892064572537666856146929704962572720}{12791111916633188643745996987200262877133548129} a^{11} - \frac{4849128435301987015787641582308072670639836797}{12791111916633188643745996987200262877133548129} a^{10} - \frac{4079385111647637698850516257162726444533924790}{12791111916633188643745996987200262877133548129} a^{9} - \frac{4625947533919885182957259292524178730728293999}{12791111916633188643745996987200262877133548129} a^{8} + \frac{5718174390350753275085225283312230589396717984}{12791111916633188643745996987200262877133548129} a^{7} + \frac{3468588823855800760655092585482961552269791888}{12791111916633188643745996987200262877133548129} a^{6} - \frac{3656612578445222128295145266236071543787840161}{12791111916633188643745996987200262877133548129} a^{5} + \frac{2971720852274079436618043385589264500228051917}{12791111916633188643745996987200262877133548129} a^{4} - \frac{4235778623024282146123826806907175860124351517}{12791111916633188643745996987200262877133548129} a^{3} - \frac{4366800776115451999863076523787851492632152800}{12791111916633188643745996987200262877133548129} a^{2} - \frac{998860647644893608992464779396280650712376989}{12791111916633188643745996987200262877133548129} a + \frac{2774082184246269772507103830469523512928584738}{12791111916633188643745996987200262877133548129}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 71048995.9777 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 15360 |
| The 90 conjugacy class representatives for t20n466 are not computed |
| Character table for t20n466 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1 |
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 | $20$ | R | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | R | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.12.11.2 | $x^{12} - 20$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ | |
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $89$ | 89.3.0.1 | $x^{3} - x + 7$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 89.3.0.1 | $x^{3} - x + 7$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.6.3.1 | $x^{6} - 178 x^{4} + 7921 x^{2} - 34543481$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |