Normalized defining polynomial
\( x^{20} - 2 x^{19} + 54 x^{18} - 97 x^{17} + 1454 x^{16} - 2178 x^{15} + 24203 x^{14} - 29499 x^{13} + \cdots + 103856419 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(9678054300479022526566826737717\) \(\medspace = 3^{10}\cdot 7^{15}\cdot 11^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.42\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $3^{1/2}7^{3/4}11^{9/10}\approx 64.51156445742586$ | ||
Ramified primes: | \(3\), \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{77}) \) | ||
$\card{ \Aut(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{21\!\cdots\!59}a^{19}-\frac{63\!\cdots\!11}{21\!\cdots\!59}a^{18}+\frac{59\!\cdots\!49}{21\!\cdots\!59}a^{17}+\frac{23\!\cdots\!00}{21\!\cdots\!59}a^{16}+\frac{81\!\cdots\!63}{21\!\cdots\!59}a^{15}+\frac{10\!\cdots\!36}{21\!\cdots\!59}a^{14}+\frac{28\!\cdots\!56}{21\!\cdots\!59}a^{13}-\frac{39\!\cdots\!65}{21\!\cdots\!59}a^{12}+\frac{67\!\cdots\!77}{21\!\cdots\!59}a^{11}+\frac{72\!\cdots\!09}{21\!\cdots\!59}a^{10}-\frac{66\!\cdots\!66}{21\!\cdots\!59}a^{9}-\frac{75\!\cdots\!92}{21\!\cdots\!59}a^{8}+\frac{10\!\cdots\!07}{21\!\cdots\!59}a^{7}-\frac{10\!\cdots\!49}{21\!\cdots\!59}a^{6}-\frac{10\!\cdots\!14}{21\!\cdots\!59}a^{5}-\frac{20\!\cdots\!23}{21\!\cdots\!59}a^{4}-\frac{60\!\cdots\!64}{21\!\cdots\!59}a^{3}+\frac{11\!\cdots\!77}{21\!\cdots\!59}a^{2}-\frac{97\!\cdots\!02}{21\!\cdots\!59}a-\frac{10\!\cdots\!09}{21\!\cdots\!59}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{81\!\cdots\!46}{21\!\cdots\!59}a^{19}+\frac{17\!\cdots\!73}{21\!\cdots\!59}a^{18}+\frac{35\!\cdots\!33}{21\!\cdots\!59}a^{17}+\frac{93\!\cdots\!42}{21\!\cdots\!59}a^{16}+\frac{78\!\cdots\!06}{21\!\cdots\!59}a^{15}+\frac{26\!\cdots\!61}{21\!\cdots\!59}a^{14}+\frac{11\!\cdots\!59}{21\!\cdots\!59}a^{13}+\frac{44\!\cdots\!65}{21\!\cdots\!59}a^{12}+\frac{10\!\cdots\!05}{21\!\cdots\!59}a^{11}+\frac{49\!\cdots\!09}{21\!\cdots\!59}a^{10}+\frac{70\!\cdots\!85}{21\!\cdots\!59}a^{9}+\frac{36\!\cdots\!11}{21\!\cdots\!59}a^{8}+\frac{30\!\cdots\!29}{21\!\cdots\!59}a^{7}+\frac{17\!\cdots\!60}{21\!\cdots\!59}a^{6}+\frac{62\!\cdots\!26}{21\!\cdots\!59}a^{5}+\frac{57\!\cdots\!67}{21\!\cdots\!59}a^{4}+\frac{44\!\cdots\!41}{21\!\cdots\!59}a^{3}+\frac{11\!\cdots\!28}{21\!\cdots\!59}a^{2}-\frac{12\!\cdots\!38}{21\!\cdots\!59}a+\frac{95\!\cdots\!04}{21\!\cdots\!59}$, $\frac{29\!\cdots\!54}{21\!\cdots\!59}a^{19}-\frac{69\!\cdots\!47}{21\!\cdots\!59}a^{18}+\frac{14\!\cdots\!91}{21\!\cdots\!59}a^{17}-\frac{32\!\cdots\!71}{21\!\cdots\!59}a^{16}+\frac{38\!\cdots\!52}{21\!\cdots\!59}a^{15}-\frac{72\!\cdots\!09}{21\!\cdots\!59}a^{14}+\frac{60\!\cdots\!83}{21\!\cdots\!59}a^{13}-\frac{97\!\cdots\!81}{21\!\cdots\!59}a^{12}+\frac{63\!\cdots\!73}{21\!\cdots\!59}a^{11}-\frac{88\!\cdots\!22}{21\!\cdots\!59}a^{10}+\frac{45\!\cdots\!82}{21\!\cdots\!59}a^{9}-\frac{58\!\cdots\!14}{21\!\cdots\!59}a^{8}+\frac{22\!\cdots\!51}{21\!\cdots\!59}a^{7}-\frac{28\!\cdots\!07}{21\!\cdots\!59}a^{6}+\frac{75\!\cdots\!43}{21\!\cdots\!59}a^{5}-\frac{90\!\cdots\!65}{21\!\cdots\!59}a^{4}+\frac{15\!\cdots\!81}{21\!\cdots\!59}a^{3}-\frac{15\!\cdots\!22}{21\!\cdots\!59}a^{2}+\frac{13\!\cdots\!70}{21\!\cdots\!59}a-\frac{95\!\cdots\!42}{21\!\cdots\!59}$, $\frac{18\!\cdots\!44}{21\!\cdots\!59}a^{19}-\frac{32\!\cdots\!94}{21\!\cdots\!59}a^{18}+\frac{93\!\cdots\!20}{21\!\cdots\!59}a^{17}-\frac{14\!\cdots\!54}{21\!\cdots\!59}a^{16}+\frac{23\!\cdots\!14}{21\!\cdots\!59}a^{15}-\frac{30\!\cdots\!02}{21\!\cdots\!59}a^{14}+\frac{37\!\cdots\!12}{21\!\cdots\!59}a^{13}-\frac{36\!\cdots\!09}{21\!\cdots\!59}a^{12}+\frac{39\!\cdots\!54}{21\!\cdots\!59}a^{11}-\frac{28\!\cdots\!18}{21\!\cdots\!59}a^{10}+\frac{28\!\cdots\!65}{21\!\cdots\!59}a^{9}-\frac{16\!\cdots\!70}{21\!\cdots\!59}a^{8}+\frac{14\!\cdots\!56}{21\!\cdots\!59}a^{7}-\frac{72\!\cdots\!86}{21\!\cdots\!59}a^{6}+\frac{49\!\cdots\!73}{21\!\cdots\!59}a^{5}-\frac{21\!\cdots\!41}{21\!\cdots\!59}a^{4}+\frac{97\!\cdots\!70}{21\!\cdots\!59}a^{3}-\frac{26\!\cdots\!86}{21\!\cdots\!59}a^{2}+\frac{80\!\cdots\!90}{21\!\cdots\!59}a+\frac{82\!\cdots\!21}{21\!\cdots\!59}$, $\frac{27\!\cdots\!64}{21\!\cdots\!59}a^{19}-\frac{54\!\cdots\!26}{21\!\cdots\!59}a^{18}+\frac{20\!\cdots\!38}{21\!\cdots\!59}a^{17}-\frac{27\!\cdots\!59}{21\!\cdots\!59}a^{16}+\frac{66\!\cdots\!32}{21\!\cdots\!59}a^{15}-\frac{68\!\cdots\!68}{21\!\cdots\!59}a^{14}+\frac{11\!\cdots\!12}{21\!\cdots\!59}a^{13}-\frac{10\!\cdots\!37}{21\!\cdots\!59}a^{12}+\frac{12\!\cdots\!14}{21\!\cdots\!59}a^{11}-\frac{10\!\cdots\!13}{21\!\cdots\!59}a^{10}+\frac{92\!\cdots\!32}{21\!\cdots\!59}a^{9}-\frac{77\!\cdots\!55}{21\!\cdots\!59}a^{8}+\frac{47\!\cdots\!66}{21\!\cdots\!59}a^{7}-\frac{38\!\cdots\!81}{21\!\cdots\!59}a^{6}+\frac{19\!\cdots\!44}{21\!\cdots\!59}a^{5}-\frac{12\!\cdots\!91}{21\!\cdots\!59}a^{4}+\frac{54\!\cdots\!70}{21\!\cdots\!59}a^{3}-\frac{24\!\cdots\!64}{21\!\cdots\!59}a^{2}+\frac{64\!\cdots\!18}{21\!\cdots\!59}a-\frac{18\!\cdots\!08}{21\!\cdots\!59}$, $\frac{16\!\cdots\!37}{21\!\cdots\!59}a^{19}-\frac{84\!\cdots\!44}{21\!\cdots\!59}a^{18}+\frac{92\!\cdots\!26}{21\!\cdots\!59}a^{17}-\frac{40\!\cdots\!42}{21\!\cdots\!59}a^{16}+\frac{24\!\cdots\!95}{21\!\cdots\!59}a^{15}-\frac{96\!\cdots\!09}{21\!\cdots\!59}a^{14}+\frac{39\!\cdots\!74}{21\!\cdots\!59}a^{13}-\frac{14\!\cdots\!38}{21\!\cdots\!59}a^{12}+\frac{41\!\cdots\!19}{21\!\cdots\!59}a^{11}-\frac{14\!\cdots\!59}{21\!\cdots\!59}a^{10}+\frac{29\!\cdots\!68}{21\!\cdots\!59}a^{9}-\frac{10\!\cdots\!57}{21\!\cdots\!59}a^{8}+\frac{14\!\cdots\!31}{21\!\cdots\!59}a^{7}-\frac{51\!\cdots\!77}{21\!\cdots\!59}a^{6}+\frac{55\!\cdots\!81}{21\!\cdots\!59}a^{5}-\frac{17\!\cdots\!94}{21\!\cdots\!59}a^{4}+\frac{13\!\cdots\!68}{21\!\cdots\!59}a^{3}-\frac{34\!\cdots\!16}{21\!\cdots\!59}a^{2}+\frac{13\!\cdots\!88}{21\!\cdots\!59}a-\frac{28\!\cdots\!65}{21\!\cdots\!59}$, $\frac{50\!\cdots\!67}{21\!\cdots\!59}a^{19}-\frac{97\!\cdots\!50}{21\!\cdots\!59}a^{18}+\frac{44\!\cdots\!31}{21\!\cdots\!59}a^{17}-\frac{48\!\cdots\!48}{21\!\cdots\!59}a^{16}+\frac{15\!\cdots\!27}{21\!\cdots\!59}a^{15}-\frac{12\!\cdots\!40}{21\!\cdots\!59}a^{14}+\frac{27\!\cdots\!51}{21\!\cdots\!59}a^{13}-\frac{18\!\cdots\!11}{21\!\cdots\!59}a^{12}+\frac{32\!\cdots\!43}{21\!\cdots\!59}a^{11}-\frac{19\!\cdots\!65}{21\!\cdots\!59}a^{10}+\frac{25\!\cdots\!93}{21\!\cdots\!59}a^{9}-\frac{13\!\cdots\!77}{21\!\cdots\!59}a^{8}+\frac{14\!\cdots\!34}{21\!\cdots\!59}a^{7}-\frac{64\!\cdots\!24}{21\!\cdots\!59}a^{6}+\frac{60\!\cdots\!61}{21\!\cdots\!59}a^{5}-\frac{20\!\cdots\!41}{21\!\cdots\!59}a^{4}+\frac{15\!\cdots\!43}{21\!\cdots\!59}a^{3}-\frac{37\!\cdots\!80}{21\!\cdots\!59}a^{2}+\frac{16\!\cdots\!00}{21\!\cdots\!59}a-\frac{26\!\cdots\!91}{21\!\cdots\!59}$, $\frac{24\!\cdots\!50}{21\!\cdots\!59}a^{19}-\frac{99\!\cdots\!48}{21\!\cdots\!59}a^{18}+\frac{11\!\cdots\!94}{21\!\cdots\!59}a^{17}-\frac{43\!\cdots\!99}{21\!\cdots\!59}a^{16}+\frac{28\!\cdots\!49}{21\!\cdots\!59}a^{15}-\frac{54\!\cdots\!01}{21\!\cdots\!59}a^{14}+\frac{42\!\cdots\!44}{21\!\cdots\!59}a^{13}-\frac{12\!\cdots\!11}{21\!\cdots\!59}a^{12}+\frac{41\!\cdots\!48}{21\!\cdots\!59}a^{11}+\frac{25\!\cdots\!05}{21\!\cdots\!59}a^{10}+\frac{28\!\cdots\!53}{21\!\cdots\!59}a^{9}+\frac{16\!\cdots\!65}{21\!\cdots\!59}a^{8}+\frac{13\!\cdots\!31}{21\!\cdots\!59}a^{7}-\frac{48\!\cdots\!82}{21\!\cdots\!59}a^{6}+\frac{40\!\cdots\!59}{21\!\cdots\!59}a^{5}-\frac{53\!\cdots\!12}{21\!\cdots\!59}a^{4}+\frac{70\!\cdots\!64}{21\!\cdots\!59}a^{3}-\frac{10\!\cdots\!68}{21\!\cdots\!59}a^{2}+\frac{52\!\cdots\!71}{21\!\cdots\!59}a-\frac{17\!\cdots\!71}{21\!\cdots\!59}$, $\frac{30\!\cdots\!91}{21\!\cdots\!59}a^{19}+\frac{40\!\cdots\!39}{21\!\cdots\!59}a^{18}+\frac{14\!\cdots\!51}{21\!\cdots\!59}a^{17}+\frac{41\!\cdots\!35}{21\!\cdots\!59}a^{16}+\frac{35\!\cdots\!89}{21\!\cdots\!59}a^{15}+\frac{20\!\cdots\!89}{21\!\cdots\!59}a^{14}+\frac{54\!\cdots\!61}{21\!\cdots\!59}a^{13}+\frac{47\!\cdots\!56}{21\!\cdots\!59}a^{12}+\frac{56\!\cdots\!84}{21\!\cdots\!59}a^{11}+\frac{62\!\cdots\!55}{21\!\cdots\!59}a^{10}+\frac{40\!\cdots\!48}{21\!\cdots\!59}a^{9}+\frac{50\!\cdots\!45}{21\!\cdots\!59}a^{8}+\frac{19\!\cdots\!03}{21\!\cdots\!59}a^{7}+\frac{24\!\cdots\!63}{21\!\cdots\!59}a^{6}+\frac{59\!\cdots\!94}{21\!\cdots\!59}a^{5}+\frac{80\!\cdots\!12}{21\!\cdots\!59}a^{4}+\frac{95\!\cdots\!06}{21\!\cdots\!59}a^{3}+\frac{16\!\cdots\!73}{21\!\cdots\!59}a^{2}+\frac{52\!\cdots\!98}{21\!\cdots\!59}a+\frac{15\!\cdots\!79}{21\!\cdots\!59}$, $\frac{57\!\cdots\!57}{21\!\cdots\!59}a^{19}-\frac{16\!\cdots\!78}{21\!\cdots\!59}a^{18}+\frac{29\!\cdots\!92}{21\!\cdots\!59}a^{17}-\frac{80\!\cdots\!74}{21\!\cdots\!59}a^{16}+\frac{74\!\cdots\!47}{21\!\cdots\!59}a^{15}-\frac{18\!\cdots\!18}{21\!\cdots\!59}a^{14}+\frac{11\!\cdots\!14}{21\!\cdots\!59}a^{13}-\frac{25\!\cdots\!56}{21\!\cdots\!59}a^{12}+\frac{11\!\cdots\!37}{21\!\cdots\!59}a^{11}-\frac{23\!\cdots\!98}{21\!\cdots\!59}a^{10}+\frac{80\!\cdots\!27}{21\!\cdots\!59}a^{9}-\frac{16\!\cdots\!49}{21\!\cdots\!59}a^{8}+\frac{37\!\cdots\!58}{21\!\cdots\!59}a^{7}-\frac{78\!\cdots\!02}{21\!\cdots\!59}a^{6}+\frac{12\!\cdots\!68}{21\!\cdots\!59}a^{5}-\frac{25\!\cdots\!45}{21\!\cdots\!59}a^{4}+\frac{22\!\cdots\!72}{21\!\cdots\!59}a^{3}-\frac{42\!\cdots\!18}{21\!\cdots\!59}a^{2}+\frac{18\!\cdots\!07}{21\!\cdots\!59}a-\frac{28\!\cdots\!35}{21\!\cdots\!59}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 7036306.946137199 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 7036306.946137199 \cdot 2}{2\cdot\sqrt{9678054300479022526566826737717}}\cr\approx \mathstrut & 0.216894946240413 \end{aligned}\] (assuming GRH)
Galois group
$C_{10}\wr C_2$ (as 20T53):
A solvable group of order 200 |
The 65 conjugacy class representatives for $C_{10}\wr C_2$ |
Character table for $C_{10}\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 4.0.33957.1, 10.0.246071287.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 20.0.661024130898095931054356037.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }^{2}$ | R | $20$ | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{10}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{10}$ | $20$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.20.10.2 | $x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$ | $2$ | $10$ | $10$ | 20T1 | $[\ ]_{2}^{10}$ |
\(7\) | 7.20.15.1 | $x^{20} + 39 x^{16} + 16 x^{15} - 344 x^{12} - 7232 x^{11} + 96 x^{10} + 4344 x^{8} + 131648 x^{7} + 38272 x^{6} + 256 x^{5} + 59536 x^{4} - 212224 x^{3} + 84096 x^{2} - 8704 x + 9328$ | $4$ | $5$ | $15$ | 20T12 | $[\ ]_{4}^{10}$ |
\(11\) | 11.10.5.1 | $x^{10} - 13310 x^{4} - 1449459$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
11.10.8.1 | $x^{10} - 77 x^{5} + 242$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |