Normalized defining polynomial
\( x^{21} - 2 x^{20} - x^{19} + 7 x^{18} - 3 x^{17} - 13 x^{16} + 14 x^{15} + 13 x^{14} - 28 x^{13} + \cdots - 1 \)
Invariants
Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(7399096241029480225646983928\) \(\medspace = 2^{3}\cdot 101\cdot 1951\cdot 7481\cdot 35629381\cdot 17609300881\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.24\) | 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: | not computed | ||
Ramified primes: | \(2\), \(101\), \(1951\), \(7481\), \(35629381\), \(17609300881\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{18497\!\cdots\!45982}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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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: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $a^{20}-2a^{19}-a^{18}+7a^{17}-3a^{16}-13a^{15}+14a^{14}+13a^{13}-28a^{12}-2a^{11}+34a^{10}-15a^{9}-25a^{8}+24a^{7}+7a^{6}-21a^{5}+6a^{4}+8a^{3}-7a^{2}+2$, $2a^{20}-5a^{19}-a^{18}+16a^{17}-10a^{16}-29a^{15}+37a^{14}+28a^{13}-70a^{12}-3a^{11}+85a^{10}-34a^{9}-67a^{8}+54a^{7}+29a^{6}-47a^{5}+3a^{4}+22a^{3}-8a^{2}-4a+3$, $2a^{20}-3a^{19}-3a^{18}+11a^{17}-22a^{15}+12a^{14}+27a^{13}-29a^{12}-19a^{11}+38a^{10}+2a^{9}-31a^{8}+8a^{7}+13a^{6}-11a^{5}-2a^{4}+2a^{3}-a^{2}-1$, $2a^{19}-5a^{18}+14a^{16}-11a^{15}-22a^{14}+34a^{13}+15a^{12}-56a^{11}+10a^{10}+57a^{9}-36a^{8}-33a^{7}+40a^{6}+4a^{5}-26a^{4}+12a^{3}+4a^{2}-6a+2$, $a^{20}+a^{19}-5a^{18}+a^{17}+14a^{16}-9a^{15}-24a^{14}+26a^{13}+26a^{12}-46a^{11}-15a^{10}+55a^{9}-4a^{8}-47a^{7}+15a^{6}+22a^{5}-20a^{4}-7a^{3}+8a^{2}-2a-3$, $2a^{20}-2a^{19}-5a^{18}+10a^{17}+7a^{16}-25a^{15}-a^{14}+41a^{13}-16a^{12}-47a^{11}+36a^{10}+35a^{9}-45a^{8}-15a^{7}+34a^{6}-6a^{5}-18a^{4}+8a^{3}+3a^{2}-5a$, $a^{20}-a^{19}-2a^{18}+4a^{17}+3a^{16}-9a^{15}-2a^{14}+14a^{13}-a^{12}-17a^{11}+4a^{10}+17a^{9}-6a^{8}-16a^{7}+6a^{6}+10a^{5}-8a^{4}-6a^{3}+6a^{2}+a-3$, $4a^{20}-7a^{19}-7a^{18}+29a^{17}-3a^{16}-63a^{15}+43a^{14}+84a^{13}-108a^{12}-63a^{11}+158a^{10}+a^{9}-153a^{8}+54a^{7}+93a^{6}-72a^{5}-27a^{4}+42a^{3}-5a^{2}-11a+4$, $2a^{19}-3a^{18}-3a^{17}+11a^{16}-22a^{14}+12a^{13}+27a^{12}-29a^{11}-19a^{10}+38a^{9}+2a^{8}-31a^{7}+9a^{6}+13a^{5}-12a^{4}+3a^{2}-2a+1$, $2a^{20}-a^{19}-6a^{18}+8a^{17}+11a^{16}-22a^{15}-10a^{14}+39a^{13}-2a^{12}-48a^{11}+19a^{10}+40a^{9}-29a^{8}-23a^{7}+22a^{6}+a^{5}-14a^{4}+2a^{3}+2a^{2}-3a-1$ (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: | \( 564425.480379 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 564425.480379 \cdot 1}{2\cdot\sqrt{7399096241029480225646983928}}\cr\approx \mathstrut & 0.629239470194 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 51090942171709440000 |
The 792 conjugacy class representatives for $S_{21}$ |
Character table for $S_{21}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 42 sibling: | 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 | R | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | $19{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $21$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.17.0.1 | $x^{17} + x^{3} + 1$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(101\) | $\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
101.4.0.1 | $x^{4} + x^{2} + 78 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
101.5.0.1 | $x^{5} + 2 x + 99$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
101.8.0.1 | $x^{8} + 4 x^{4} + 76 x^{3} + 29 x^{2} + 24 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
\(1951\) | $\Q_{1951}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(7481\) | $\Q_{7481}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(35629381\) | $\Q_{35629381}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(17609300881\) | $\Q_{17609300881}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17609300881}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |