Normalized defining polynomial
\( x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 55 x^{16} - 897 x^{15} - 410 x^{14} + 2525 x^{13} + \cdots - 1 \)
Invariants
Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[17, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(38289048631639269472806869479351853\)
\(\medspace = 7^{14}\cdot 13\cdot 97\cdot 8914877\cdot 5021940620101\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.34\) | 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: | $7^{2/3}13^{1/2}97^{1/2}8914877^{1/2}5021940620101^{1/2}\approx 869460188568.4381$ | ||
Ramified primes: |
\(7\), \(13\), \(97\), \(8914877\), \(5021940620101\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{56454\!\cdots\!89597}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $18$ | 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: |
$a^{19}-a^{18}-20a^{17}+17a^{16}+169a^{15}-114a^{14}-783a^{13}+373a^{12}+2152a^{11}-568a^{10}-3525a^{9}+186a^{8}+3256a^{7}+438a^{6}-1499a^{5}-403a^{4}+304a^{3}+107a^{2}-22a-7$, $a^{20}+3a^{19}-24a^{18}-64a^{17}+239a^{16}+580a^{15}-1274a^{14}-2893a^{13}+3892a^{12}+8575a^{11}-6706a^{10}-15156a^{9}+5823a^{8}+15153a^{7}-1687a^{6}-7670a^{5}-335a^{4}+1763a^{3}+226a^{2}-142a-26$, $a$, $a+1$, $14a^{20}-5a^{19}-292a^{18}+62a^{17}+2578a^{16}-146a^{15}-12477a^{14}-1315a^{13}+35703a^{12}+9529a^{11}-60428a^{10}-25351a^{9}+56848a^{8}+31585a^{7}-25797a^{6}-17485a^{5}+4620a^{4}+4098a^{3}-63a^{2}-326a-40$, $a^{20}-21a^{18}-3a^{17}+186a^{16}+55a^{15}-897a^{14}-410a^{13}+2525a^{12}+1584a^{11}-4093a^{10}-3339a^{9}+3442a^{8}+3694a^{7}-1060a^{6}-1902a^{5}-106a^{4}+411a^{3}+98a^{2}-29a-11$, $a^{20}-21a^{18}-3a^{17}+186a^{16}+55a^{15}-897a^{14}-410a^{13}+2525a^{12}+1584a^{11}-4093a^{10}-3339a^{9}+3442a^{8}+3694a^{7}-1060a^{6}-1902a^{5}-106a^{4}+410a^{3}+98a^{2}-26a-10$, $a^{20}+2a^{19}-23a^{18}-44a^{17}+222a^{16}+411a^{15}-1160a^{14}-2110a^{13}+3519a^{12}+6423a^{11}-6138a^{10}-11631a^{9}+5637a^{8}+11897a^{7}-2125a^{6}-6171a^{5}+68a^{4}+1459a^{3}+119a^{2}-121a-18$, $2a^{20}-2a^{19}-41a^{18}+36a^{17}+356a^{16}-263a^{15}-1700a^{14}+994a^{13}+4839a^{12}-2045a^{11}-8292a^{10}+2195a^{9}+8203a^{8}-1065a^{7}-4269a^{6}+174a^{5}+1049a^{4}+21a^{3}-95a^{2}-8a+2$, $a^{20}-21a^{18}-3a^{17}+186a^{16}+55a^{15}-897a^{14}-410a^{13}+2525a^{12}+1584a^{11}-4093a^{10}-3339a^{9}+3442a^{8}+3694a^{7}-1060a^{6}-1902a^{5}-106a^{4}+411a^{3}+97a^{2}-28a-10$, $8a^{20}-2a^{19}-167a^{18}+17a^{17}+1474a^{16}+85a^{15}-7117a^{14}-1600a^{13}+20237a^{12}+7995a^{11}-33760a^{10}-19094a^{9}+30689a^{8}+22854a^{7}-12613a^{6}-12656a^{5}+1464a^{4}+3076a^{3}+254a^{2}-268a-50$, $a^{20}-a^{19}-20a^{18}+17a^{17}+169a^{16}-114a^{15}-783a^{14}+373a^{13}+2152a^{12}-568a^{11}-3525a^{10}+186a^{9}+3256a^{8}+438a^{7}-1499a^{6}-403a^{5}+304a^{4}+107a^{3}-22a^{2}-7a+1$, $3a^{20}-6a^{19}-58a^{18}+114a^{17}+472a^{16}-902a^{15}-2106a^{14}+3850a^{13}+5624a^{12}-9604a^{11}-9204a^{10}+14198a^{9}+9078a^{8}-12086a^{7}-5141a^{6}+5538a^{5}+1583a^{4}-1219a^{3}-250a^{2}+94a+18$, $55a^{20}-22a^{19}-1143a^{18}+289a^{17}+10050a^{16}-940a^{15}-48411a^{14}-3561a^{13}+137741a^{12}+33294a^{11}-231345a^{10}-93219a^{9}+214899a^{8}+118449a^{7}-94808a^{6}-66090a^{5}+15595a^{4}+15562a^{3}+168a^{2}-1251a-169$, $a^{19}-a^{18}-20a^{17}+17a^{16}+169a^{15}-114a^{14}-783a^{13}+373a^{12}+2152a^{11}-568a^{10}-3525a^{9}+186a^{8}+3256a^{7}+438a^{6}-1499a^{5}-403a^{4}+303a^{3}+107a^{2}-19a-7$, $2a^{19}-2a^{18}-41a^{17}+36a^{16}+356a^{15}-263a^{14}-1700a^{13}+994a^{12}+4839a^{11}-2045a^{10}-8292a^{9}+2195a^{8}+8203a^{7}-1065a^{6}-4269a^{5}+174a^{4}+1049a^{3}+21a^{2}-94a-8$, $33a^{20}-20a^{19}-680a^{18}+312a^{17}+5932a^{16}-1761a^{15}-28397a^{14}+3550a^{13}+80570a^{12}+3886a^{11}-135849a^{10}-28627a^{9}+128492a^{8}+44610a^{7}-59893a^{6}-26502a^{5}+11665a^{4}+6348a^{3}-454a^{2}-517a-58$, $45a^{20}-25a^{19}-929a^{18}+379a^{17}+8117a^{16}-1998a^{15}-38894a^{14}+2907a^{13}+110323a^{12}+10852a^{11}-185520a^{10}-48678a^{9}+174094a^{8}+70520a^{7}-79384a^{6}-41251a^{5}+14461a^{4}+9920a^{3}-273a^{2}-816a-107$
| 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: | \( 14921510518.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{2}\cdot 14921510518.5 \cdot 1}{2\cdot\sqrt{38289048631639269472806869479351853}}\cr\approx \mathstrut & 0.197294586501 \end{aligned}\] (assuming GRH)
Galois group
$S_7\wr C_3$ (as 21T159):
A non-solvable group of order 384072192000 |
The 1165 conjugacy class representatives for $S_7\wr C_3$ |
Character table for $S_7\wr C_3$ |
Intermediate fields
\(\Q(\zeta_{7})^+\) |
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 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $18{,}\,{\href{/padicField/2.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | $15{,}\,{\href{/padicField/11.6.0.1}{6} }$ | R | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $21$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\)
| Deg $21$ | $3$ | $7$ | $14$ | |||
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.7.0.1 | $x^{7} + 3 x + 11$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(97\)
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
97.4.0.1 | $x^{4} + 6 x^{2} + 80 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
97.4.0.1 | $x^{4} + 6 x^{2} + 80 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
97.5.0.1 | $x^{5} + 3 x + 92$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(8914877\)
| $\Q_{8914877}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{8914877}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{8914877}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(5021940620101\)
| $\Q_{5021940620101}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5021940620101}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5021940620101}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5021940620101}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5021940620101}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5021940620101}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{5021940620101}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |