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: | \(75375354289024738220986584068\) \(\medspace = 2^{2}\cdot 11\cdot 29\cdot 1447\cdot 2579\cdot 4423\cdot 3578836676945557\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(23.72\) | 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))
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Galois root discriminant: | $2^{2/3}11^{1/2}29^{1/2}1447^{1/2}2579^{1/2}4423^{1/2}3578836676945557^{1/2}\approx 217907085144134.4$ | ||
Ramified primes: | \(2\), \(11\), \(29\), \(1447\), \(2579\), \(4423\), \(3578836676945557\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{18843\!\cdots\!46017}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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Fundamental units: | $a$, $a^{20}-a^{19}-2a^{18}+4a^{17}+3a^{16}-9a^{15}-2a^{14}+14a^{13}-a^{12}-17a^{11}+4a^{10}+17a^{9}-6a^{8}-14a^{7}+6a^{6}+9a^{5}-7a^{4}-7a^{3}+4a^{2}+a-2$, $a^{20}-2a^{19}-a^{18}+7a^{17}-3a^{16}-13a^{15}+14a^{14}+13a^{13}-28a^{12}-2a^{11}+34a^{10}-15a^{9}-25a^{8}+25a^{7}+7a^{6}-22a^{5}+6a^{4}+8a^{3}-7a^{2}+a+1$, $a^{19}-2a^{18}-a^{17}+7a^{16}-3a^{15}-13a^{14}+13a^{13}+14a^{12}-26a^{11}-6a^{10}+32a^{9}-7a^{8}-25a^{7}+15a^{6}+11a^{5}-13a^{4}+4a^{2}-2a$, $2a^{20}-3a^{19}-4a^{18}+12a^{17}+2a^{16}-26a^{15}+10a^{14}+35a^{13}-30a^{12}-28a^{11}+44a^{10}+6a^{9}-39a^{8}+15a^{7}+18a^{6}-23a^{5}+12a^{3}-5a^{2}-2a+1$, $a^{18}-2a^{17}-a^{16}+6a^{15}-a^{14}-12a^{13}+8a^{12}+14a^{11}-16a^{10}-10a^{9}+20a^{8}+a^{7}-15a^{6}+5a^{5}+6a^{4}-7a^{3}+a^{2}+2a-1$, $2a^{20}-5a^{19}+14a^{17}-11a^{16}-22a^{15}+34a^{14}+15a^{13}-56a^{12}+11a^{11}+56a^{10}-38a^{9}-30a^{8}+44a^{7}-2a^{6}-29a^{5}+17a^{4}+3a^{3}-7a^{2}+3a$, $a^{20}-3a^{19}-a^{18}+10a^{17}-4a^{16}-21a^{15}+18a^{14}+28a^{13}-38a^{12}-22a^{11}+52a^{10}+5a^{9}-47a^{8}+11a^{7}+27a^{6}-17a^{5}-4a^{4}+11a^{3}-3a^{2}-2a+2$, $a^{20}-a^{19}-2a^{18}+4a^{17}+3a^{16}-9a^{15}-2a^{14}+14a^{13}-2a^{12}-16a^{11}+5a^{10}+14a^{9}-7a^{8}-9a^{7}+5a^{6}+4a^{5}-4a^{4}-3a^{3}+a^{2}+a$, $a^{20}-3a^{19}+10a^{17}-9a^{16}-16a^{15}+28a^{14}+11a^{13}-49a^{12}+11a^{11}+53a^{10}-37a^{9}-33a^{8}+46a^{7}+3a^{6}-32a^{5}+14a^{4}+9a^{3}-10a^{2}+4a$ (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)]
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Regulator: | \( 2395762.98687 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 2395762.98687 \cdot 1}{2\cdot\sqrt{75375354289024738220986584068}}\cr\approx \mathstrut & 0.836811907446 \end{aligned}\] (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 | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | $18{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $19{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.7.0.1 | $x^{7} + 4 x + 9$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
11.11.0.1 | $x^{11} + 10 x + 9$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(29\) | 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
29.13.0.1 | $x^{13} + 7 x + 27$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
\(1447\) | $\Q_{1447}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1447}$ | $x$ | $1$ | $1$ | $0$ | 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 $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(2579\) | 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(4423\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(3578836676945557\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |