Normalized defining polynomial
\( x^{22} - 11 x^{20} + 33 x^{18} - 11 x^{16} - 22 x^{14} - 22 x^{12} - 80 x^{11} - 22 x^{10} - 22 x^{8} + \cdots + 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(24111231168499845791571893752356143104\) \(\medspace = 2^{20}\cdot 7^{10}\cdot 11^{22}\) | 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: | \(50.03\) | 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: | $2^{10/11}7^{1/2}11^{119/110}\approx 66.49824652345667$ | ||
Ramified primes: | \(2\), \(7\), \(11\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{3}{8}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{8}a^{18}-\frac{1}{4}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{19}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{8}a^{20}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{48}a^{21}+\frac{1}{48}a^{20}+\frac{1}{24}a^{19}+\frac{1}{24}a^{18}-\frac{1}{48}a^{17}-\frac{1}{48}a^{16}+\frac{1}{24}a^{13}+\frac{1}{24}a^{12}+\frac{1}{12}a^{11}+\frac{1}{6}a^{10}-\frac{1}{24}a^{9}+\frac{5}{24}a^{8}-\frac{11}{48}a^{5}-\frac{11}{48}a^{4}+\frac{5}{24}a^{3}+\frac{11}{24}a^{2}+\frac{11}{48}a-\frac{1}{48}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | 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: | $a^{21}-11a^{19}+33a^{17}-11a^{15}-22a^{13}-22a^{11}-80a^{10}-22a^{9}-22a^{7}-11a^{5}+33a^{3}-11a$, $\frac{97}{24}a^{21}-\frac{35}{6}a^{20}-\frac{527}{12}a^{19}+\frac{377}{6}a^{18}+\frac{3059}{24}a^{17}-\frac{1069}{6}a^{16}-27a^{15}+\frac{95}{4}a^{14}-\frac{1127}{12}a^{13}+\frac{797}{6}a^{12}-\frac{301}{3}a^{11}-\frac{1967}{12}a^{10}+\frac{4385}{12}a^{9}+\frac{362}{3}a^{8}+\frac{9}{2}a^{7}+\frac{649}{4}a^{6}-\frac{767}{24}a^{5}+\frac{305}{3}a^{4}+\frac{1577}{12}a^{3}-\frac{2035}{12}a^{2}-\frac{601}{24}a+\frac{155}{6}$, $23a^{21}-2a^{20}-\frac{993}{4}a^{19}+\frac{87}{4}a^{18}+\frac{2831}{4}a^{17}-\frac{253}{4}a^{16}-107a^{15}+\frac{55}{4}a^{14}-528a^{13}+\frac{187}{4}a^{12}-\frac{2453}{4}a^{11}-\frac{3581}{2}a^{10}-\frac{1901}{4}a^{9}-\frac{661}{2}a^{8}-618a^{7}-\frac{123}{4}a^{6}-385a^{5}+\frac{37}{4}a^{4}+\frac{1359}{2}a^{3}-\frac{261}{4}a^{2}-\frac{227}{2}a+\frac{51}{4}$, $2a^{21}+\frac{21}{4}a^{20}-\frac{87}{4}a^{19}-\frac{227}{4}a^{18}+\frac{253}{4}a^{17}+\frac{325}{2}a^{16}-\frac{55}{4}a^{15}-\frac{55}{2}a^{14}-\frac{187}{4}a^{13}-\frac{473}{4}a^{12}-\frac{99}{2}a^{11}-\frac{1201}{4}a^{10}-\frac{939}{2}a^{9}-163a^{8}-\frac{519}{4}a^{7}-143a^{6}-\frac{183}{4}a^{5}-\frac{171}{2}a^{4}+\frac{243}{4}a^{3}+\frac{315}{2}a^{2}-\frac{57}{4}a-28$, $\frac{97}{12}a^{21}-\frac{115}{24}a^{20}-\frac{2093}{24}a^{19}+\frac{1243}{24}a^{18}+\frac{2981}{12}a^{17}-\frac{3557}{24}a^{16}-\frac{75}{2}a^{15}+\frac{97}{4}a^{14}-\frac{547}{3}a^{13}+\frac{1325}{12}a^{12}-\frac{656}{3}a^{11}-\frac{6241}{12}a^{10}+\frac{1921}{12}a^{9}-\frac{55}{6}a^{8}-156a^{7}+\frac{385}{4}a^{6}-\frac{1523}{12}a^{5}+\frac{1685}{24}a^{4}+\frac{5729}{24}a^{3}-\frac{3541}{24}a^{2}-\frac{118}{3}a+\frac{661}{24}$, $\frac{25}{24}a^{21}-\frac{31}{12}a^{20}-\frac{137}{12}a^{19}+\frac{167}{6}a^{18}+\frac{815}{24}a^{17}-\frac{947}{12}a^{16}-\frac{21}{2}a^{15}+10a^{14}-\frac{269}{12}a^{13}+\frac{184}{3}a^{12}-\frac{70}{3}a^{11}-\frac{44}{3}a^{10}+\frac{2189}{12}a^{9}+\frac{403}{6}a^{8}+\frac{43}{2}a^{7}+69a^{6}-\frac{191}{24}a^{5}+\frac{503}{12}a^{4}+\frac{449}{12}a^{3}-\frac{437}{6}a^{2}-\frac{145}{24}a+\frac{145}{12}$, $\frac{43}{4}a^{21}-\frac{73}{8}a^{20}-\frac{467}{4}a^{19}+\frac{197}{2}a^{18}+\frac{677}{2}a^{17}-\frac{2249}{8}a^{16}-\frac{285}{4}a^{15}+45a^{14}-\frac{989}{4}a^{13}+\frac{819}{4}a^{12}-\frac{1067}{4}a^{11}-622a^{10}+\frac{1841}{4}a^{9}+\frac{539}{4}a^{8}-\frac{523}{4}a^{7}+246a^{6}-\frac{247}{2}a^{5}+\frac{1219}{8}a^{4}+\frac{695}{2}a^{3}-\frac{547}{2}a^{2}-\frac{267}{4}a+\frac{363}{8}$, $\frac{93}{8}a^{21}-\frac{35}{8}a^{20}-\frac{1003}{8}a^{19}+\frac{95}{2}a^{18}+\frac{713}{2}a^{17}-\frac{1101}{8}a^{16}-\frac{99}{2}a^{15}+29a^{14}-\frac{1083}{4}a^{13}+\frac{391}{4}a^{12}-\frac{1265}{4}a^{11}-816a^{10}+35a^{9}-\frac{333}{4}a^{8}-269a^{7}+76a^{6}-\frac{1507}{8}a^{5}+\frac{445}{8}a^{4}+\frac{2705}{8}a^{3}-\frac{289}{2}a^{2}-56a+\frac{215}{8}$, $\frac{33}{2}a^{21}-\frac{13}{4}a^{20}-178a^{19}+\frac{141}{4}a^{18}+\frac{2027}{4}a^{17}-\frac{407}{4}a^{16}-\frac{297}{4}a^{15}+\frac{77}{4}a^{14}-\frac{753}{2}a^{13}+77a^{12}-451a^{11}-\frac{2475}{2}a^{10}-\frac{753}{4}a^{9}-\frac{791}{4}a^{8}-\frac{1677}{4}a^{7}+\frac{101}{4}a^{6}-\frac{561}{2}a^{5}+\frac{151}{4}a^{4}+\frac{969}{2}a^{3}-\frac{457}{4}a^{2}-82a+22$, $\frac{13}{8}a^{21}+\frac{1}{8}a^{20}-\frac{35}{2}a^{19}-a^{18}+\frac{99}{2}a^{17}+\frac{1}{4}a^{16}-\frac{11}{2}a^{15}+\frac{33}{4}a^{14}-\frac{159}{4}a^{13}+\frac{5}{4}a^{12}-\frac{181}{4}a^{11}-\frac{285}{2}a^{10}-\frac{205}{4}a^{9}-\frac{197}{4}a^{8}-\frac{143}{2}a^{7}-\frac{77}{4}a^{6}-\frac{383}{8}a^{5}-\frac{91}{8}a^{4}+\frac{159}{4}a^{3}-\frac{15}{2}a^{2}-\frac{25}{4}a+3$, $\frac{601}{24}a^{21}-\frac{151}{12}a^{20}-\frac{3245}{12}a^{19}+\frac{815}{6}a^{18}+\frac{18527}{24}a^{17}-\frac{4649}{12}a^{16}-\frac{481}{4}a^{15}+\frac{119}{2}a^{14}-\frac{3451}{6}a^{13}+\frac{1721}{6}a^{12}-\frac{7969}{12}a^{11}-\frac{9991}{6}a^{10}+\frac{959}{3}a^{9}-\frac{347}{6}a^{8}-\frac{1935}{4}a^{7}+\frac{527}{2}a^{6}-\frac{8909}{24}a^{5}+\frac{2285}{12}a^{4}+\frac{4495}{6}a^{3}-\frac{1123}{3}a^{2}-\frac{2935}{24}a+\frac{739}{12}$, $20a^{21}+11a^{20}-\frac{433}{2}a^{19}-119a^{18}+\frac{4977}{8}a^{17}+\frac{2731}{8}a^{16}-\frac{445}{4}a^{15}-59a^{14}-\frac{1837}{4}a^{13}-253a^{12}-\frac{2079}{4}a^{11}-1886a^{10}-1412a^{9}-\frac{2303}{4}a^{8}-\frac{2777}{4}a^{7}-\frac{693}{2}a^{6}-\frac{1373}{4}a^{5}-\frac{371}{2}a^{4}+\frac{2387}{4}a^{3}+\frac{659}{2}a^{2}-\frac{925}{8}a-\frac{489}{8}$, $\frac{269}{4}a^{21}-\frac{119}{4}a^{20}-\frac{2905}{4}a^{19}+\frac{2571}{8}a^{18}+\frac{4147}{2}a^{17}-\frac{3673}{4}a^{16}-324a^{15}+\frac{291}{2}a^{14}-\frac{6171}{4}a^{13}+\frac{2739}{4}a^{12}-\frac{7161}{4}a^{11}-4592a^{10}+\frac{1075}{2}a^{9}-270a^{8}-1378a^{7}+597a^{6}-1018a^{5}+448a^{4}+\frac{4035}{2}a^{3}-\frac{7127}{8}a^{2}-\frac{671}{2}a+\frac{593}{4}$, $\frac{331}{24}a^{21}+\frac{103}{6}a^{20}-\frac{3541}{24}a^{19}-\frac{554}{3}a^{18}+\frac{4927}{12}a^{17}+\frac{6259}{12}a^{16}-28a^{15}-\frac{245}{4}a^{14}-\frac{1867}{6}a^{13}-\frac{2371}{6}a^{12}-\frac{1189}{3}a^{11}-\frac{4730}{3}a^{10}-\frac{5392}{3}a^{9}-\frac{9901}{12}a^{8}-\frac{1539}{2}a^{7}-\frac{2385}{4}a^{6}-\frac{8699}{24}a^{5}-\frac{2051}{6}a^{4}+\frac{8425}{24}a^{3}+\frac{2881}{6}a^{2}-\frac{521}{12}a-\frac{215}{3}$, $\frac{1757}{12}a^{21}-\frac{1817}{24}a^{20}-\frac{37993}{24}a^{19}+\frac{4913}{6}a^{18}+\frac{54409}{12}a^{17}-\frac{28169}{12}a^{16}-759a^{15}+400a^{14}-\frac{40313}{12}a^{13}+\frac{5221}{3}a^{12}-\frac{46279}{12}a^{11}-\frac{29195}{3}a^{10}+\frac{25271}{12}a^{9}-\frac{2023}{12}a^{8}-2843a^{7}+\frac{3247}{2}a^{6}-\frac{6451}{3}a^{5}+\frac{27205}{24}a^{4}+\frac{106351}{24}a^{3}-\frac{6856}{3}a^{2}-\frac{2336}{3}a+\frac{2447}{6}$ (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: | \( 636267844897 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 636267844897 \cdot 1}{2\cdot\sqrt{24111231168499845791571893752356143104}}\cr\approx \mathstrut & 4.08206065430513 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.F_{11}$ (as 22T34):
A solvable group of order 112640 |
The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
Character table for $C_2^{10}.F_{11}$ |
Intermediate fields
11.11.4910318845910094848.1 |
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 | R | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | $20{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | $20{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.11.10.1 | $x^{11} + 2$ | $11$ | $1$ | $10$ | $F_{11}$ | $[\ ]_{11}^{10}$ |
2.11.10.1 | $x^{11} + 2$ | $11$ | $1$ | $10$ | $F_{11}$ | $[\ ]_{11}^{10}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
7.20.10.1 | $x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
\(11\) | 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |