Normalized defining polynomial
\( x^{22} - 11 x^{21} + 65 x^{20} - 265 x^{19} + 825 x^{18} - 2067 x^{17} + 4301 x^{16} - 7582 x^{15} + \cdots + 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(178901193768541138691065033\) \(\medspace = 149\cdot 33797\cdot 64661^{2}\cdot 92179^{2}\) | 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: | \(15.61\) | 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: | $149^{1/2}33797^{1/2}64661^{1/2}92179^{1/2}\approx 173248472.68320495$ | ||
Ramified primes: | \(149\), \(33797\), \(64661\), \(92179\) | 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: | $\Q(\sqrt{5035753}$) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $a^{2}-a+1$, $2a^{20}-20a^{19}+107a^{18}-393a^{17}+1096a^{16}-2444a^{15}+4492a^{14}-6932a^{13}+9080a^{12}-10150a^{11}+9691a^{10}-7877a^{9}+5413a^{8}-3114a^{7}+1492a^{6}-606a^{5}+232a^{4}-98a^{3}+45a^{2}-16a+2$, $a^{18}-9a^{17}+43a^{16}-140a^{15}+344a^{14}-672a^{13}+1076a^{12}-1438a^{11}+1620a^{10}-1544a^{9}+1243a^{8}-840a^{7}+473a^{6}-221a^{5}+90a^{4}-36a^{3}+17a^{2}-7a+2$, $a^{20}-10a^{19}+54a^{18}-201a^{17}+569a^{16}-1288a^{15}+2400a^{14}-3746a^{13}+4946a^{12}-5548a^{11}+5284a^{10}-4250a^{9}+2858a^{8}-1584a^{7}+718a^{6}-275a^{5}+107a^{4}-52a^{3}+26a^{2}-9a-1$, $2a^{20}-20a^{19}+106a^{18}-384a^{17}+1052a^{16}-2296a^{15}+4115a^{14}-6169a^{13}+7819a^{12}-8421a^{11}+7709a^{10}-5975a^{9}+3893a^{8}-2115a^{7}+962a^{6}-384a^{5}+156a^{4}-72a^{3}+31a^{2}-9a$, $a^{19}-9a^{18}+43a^{17}-140a^{16}+342a^{15}-658a^{14}+1023a^{13}-1302a^{12}+1360a^{11}-1157a^{10}+785a^{9}-406a^{8}+146a^{7}-29a^{6}+3a^{5}-5a^{4}+4a^{3}-a^{2}-3a$, $a^{21}-11a^{20}+63a^{19}-245a^{18}+719a^{17}-1682a^{16}+3240a^{15}-5244a^{14}+7222a^{13}-8524a^{12}+8647a^{11}-7532a^{10}+5609a^{9}-3546a^{8}+1892a^{7}-858a^{6}+349a^{5}-144a^{4}+65a^{3}-26a^{2}+7a-1$, $2a^{21}-20a^{20}+107a^{19}-394a^{18}+1104a^{17}-2480a^{16}+4606a^{15}-7211a^{14}+9631a^{13}-11049a^{12}+10918a^{11}-9285a^{10}+6772a^{9}-4213a^{8}+2231a^{7}-1017a^{6}+421a^{5}-175a^{4}+74a^{3}-29a^{2}+6a$, $a^{20}-10a^{19}+53a^{18}-191a^{17}+518a^{16}-1113a^{15}+1952a^{14}-2843a^{13}+3469a^{12}-3553a^{11}+3040a^{10}-2146a^{9}+1223a^{8}-544a^{7}+185a^{6}-55a^{5}+27a^{4}-17a^{3}+7a^{2}-1$, $a^{17}-9a^{16}+43a^{15}-139a^{14}+336a^{13}-639a^{12}+984a^{11}-1247a^{10}+1310a^{9}-1141a^{8}+819a^{7}-479a^{6}+227a^{5}-89a^{4}+34a^{3}-16a^{2}+8a-2$, $a^{18}-9a^{17}+43a^{16}-140a^{15}+343a^{14}-665a^{13}+1049a^{12}-1367a^{11}+1480a^{10}-1328a^{9}+977a^{8}-577a^{7}+266a^{6}-94a^{5}+31a^{4}-15a^{3}+9a^{2}-3a-1$ (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: | \( 11377.8928745 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 11377.8928745 \cdot 1}{2\cdot\sqrt{178901193768541138691065033}}\cr\approx \mathstrut & 0.163148806257 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
A non-solvable group of order 81749606400 |
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ |
Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
11.1.5960386319.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 sibling: | data not computed |
Degree 44 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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | $22$ |
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 |
---|---|---|---|---|---|---|---|
\(149\) | $\Q_{149}$ | $x + 147$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{149}$ | $x + 147$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
149.2.1.2 | $x^{2} + 298$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
149.3.0.1 | $x^{3} + 3 x + 147$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
149.3.0.1 | $x^{3} + 3 x + 147$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
149.12.0.1 | $x^{12} + 121 x^{6} + 91 x^{5} + 52 x^{4} + 9 x^{3} + 104 x^{2} + 110 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(33797\) | $\Q_{33797}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{33797}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(64661\) | $\Q_{64661}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{64661}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(92179\) | Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |