Normalized defining polynomial
\( x^{20} - 10 x^{19} + 38 x^{18} - 39 x^{17} - 264 x^{16} + 1265 x^{15} - 2152 x^{14} - 596 x^{13} + 11186 x^{12} - 22584 x^{11} + 9350 x^{10} + 39944 x^{9} - 77472 x^{8} + 44190 x^{7} + 31876 x^{6} - 105543 x^{5} + 102557 x^{4} + 15219 x^{3} - 70483 x^{2} + 28531 x - 1919 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(767371757239298309394737548828125=3^{10}\cdot 5^{13}\cdot 239^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $44.08$ | 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: | $3, 5, 239$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{4}{11} a^{14} - \frac{5}{11} a^{13} - \frac{2}{11} a^{12} + \frac{3}{11} a^{10} - \frac{3}{11} a^{8} - \frac{5}{11} a^{7} + \frac{5}{11} a^{6} + \frac{3}{11} a^{5} - \frac{5}{11} a^{4} + \frac{1}{11} a^{3} + \frac{1}{11} a^{2} + \frac{2}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{16} + \frac{1}{11} a^{14} - \frac{4}{11} a^{13} - \frac{3}{11} a^{12} + \frac{3}{11} a^{11} - \frac{1}{11} a^{10} - \frac{3}{11} a^{9} - \frac{4}{11} a^{8} + \frac{3}{11} a^{7} + \frac{5}{11} a^{6} + \frac{5}{11} a^{5} - \frac{1}{11} a^{4} - \frac{3}{11} a^{3} - \frac{2}{11} a^{2} - \frac{4}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{17} + \frac{3}{11} a^{14} + \frac{2}{11} a^{13} + \frac{5}{11} a^{12} - \frac{1}{11} a^{11} + \frac{5}{11} a^{10} - \frac{4}{11} a^{9} - \frac{5}{11} a^{8} - \frac{1}{11} a^{7} - \frac{4}{11} a^{5} + \frac{2}{11} a^{4} - \frac{3}{11} a^{3} - \frac{5}{11} a^{2} + \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{121} a^{18} + \frac{5}{121} a^{17} - \frac{1}{121} a^{16} + \frac{3}{121} a^{15} - \frac{28}{121} a^{14} + \frac{41}{121} a^{13} - \frac{50}{121} a^{12} + \frac{19}{121} a^{11} - \frac{5}{11} a^{10} + \frac{4}{11} a^{9} + \frac{4}{11} a^{8} + \frac{58}{121} a^{7} + \frac{57}{121} a^{6} + \frac{54}{121} a^{5} - \frac{58}{121} a^{4} - \frac{39}{121} a^{3} + \frac{25}{121} a^{2} + \frac{42}{121} a + \frac{18}{121}$, $\frac{1}{51520476945949127753420711287676379301706017} a^{19} + \frac{79505404334259336909241456709812865971496}{51520476945949127753420711287676379301706017} a^{18} + \frac{69670626364746842654440255259787513126691}{4683679722359011613947337389788761754700547} a^{17} + \frac{333480059893601970194305101769065989896774}{51520476945949127753420711287676379301706017} a^{16} - \frac{1555813109843318169419688877087646922144822}{51520476945949127753420711287676379301706017} a^{15} + \frac{20365801819209197985195289378122989414805378}{51520476945949127753420711287676379301706017} a^{14} - \frac{180147612839277555359922718660775835066051}{4683679722359011613947337389788761754700547} a^{13} + \frac{16965270147659422729689799560010299168795036}{51520476945949127753420711287676379301706017} a^{12} - \frac{18576512524164572624433991976047609479080764}{51520476945949127753420711287676379301706017} a^{11} - \frac{1338713998050332067099248235194032792341779}{4683679722359011613947337389788761754700547} a^{10} + \frac{192793849643312064685858826268705464728434}{425789065669001055813394308162614704972777} a^{9} + \frac{24724924959102109752069020214715382087287970}{51520476945949127753420711287676379301706017} a^{8} - \frac{13409750468931814342533927154048590395418248}{51520476945949127753420711287676379301706017} a^{7} + \frac{21035831221393052673048199661841837032068467}{51520476945949127753420711287676379301706017} a^{6} - \frac{12078546982595212281945431212313061346090090}{51520476945949127753420711287676379301706017} a^{5} - \frac{514932890106710300152728528538565550771750}{4683679722359011613947337389788761754700547} a^{4} + \frac{9981074687853117350955818342082906497983609}{51520476945949127753420711287676379301706017} a^{3} - \frac{22193074274987571783782459610139049798138056}{51520476945949127753420711287676379301706017} a^{2} - \frac{273252502410792279987484676015215678863048}{4683679722359011613947337389788761754700547} a + \frac{1331937422634593002438334892787671702869871}{51520476945949127753420711287676379301706017}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 2284207916.87 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 640 |
| The 40 conjugacy class representatives for t20n144 |
| Character table for t20n144 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1 |
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 | $20$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ | $20$ | $20$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.8.6.1 | $x^{8} + 9 x^{4} + 36$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 239 | Data not computed | ||||||