Normalized defining polynomial
\( x^{20} + x^{18} + 3x^{16} + 7x^{14} - 3x^{12} - 4x^{10} - 13x^{8} - 25x^{6} + 7x^{4} + 4x^{2} - 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-13167225195265942636954624\) \(\medspace = -\,2^{10}\cdot 11^{16}\cdot 23^{4}\) | 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: | \(18.03\) | 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: | $2^{31/16}11^{4/5}23^{1/2}\approx 125.0903140152616$ | ||
Ramified primes: | \(2\), \(11\), \(23\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{5722}a^{18}-\frac{275}{2861}a^{16}-\frac{213}{5722}a^{14}-\frac{1}{2}a^{13}+\frac{69}{5722}a^{12}-\frac{1}{2}a^{11}-\frac{829}{5722}a^{10}-\frac{1}{2}a^{9}-\frac{985}{5722}a^{8}-\frac{434}{2861}a^{6}+\frac{228}{2861}a^{4}-\frac{1171}{2861}a^{2}-\frac{1}{2}a-\frac{1363}{2861}$, $\frac{1}{5722}a^{19}-\frac{275}{2861}a^{17}-\frac{213}{5722}a^{15}-\frac{1}{2}a^{14}+\frac{69}{5722}a^{13}-\frac{1}{2}a^{12}-\frac{829}{5722}a^{11}-\frac{1}{2}a^{10}-\frac{985}{5722}a^{9}-\frac{434}{2861}a^{7}+\frac{228}{2861}a^{5}-\frac{1171}{2861}a^{3}-\frac{1}{2}a^{2}-\frac{1363}{2861}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $a^{19}+a^{17}+3a^{15}+7a^{13}-3a^{11}-4a^{9}-13a^{7}-25a^{5}+7a^{3}+4a$, $\frac{6945}{2861}a^{19}+\frac{8268}{2861}a^{17}+\frac{22740}{2861}a^{15}+\frac{52916}{2861}a^{13}-\frac{9656}{2861}a^{11}-\frac{28784}{2861}a^{9}-\frac{97407}{2861}a^{7}-\frac{191894}{2861}a^{5}+\frac{5317}{2861}a^{3}+\frac{27777}{2861}a$, $\frac{3372}{2861}a^{18}+\frac{5050}{2861}a^{16}+\frac{11319}{2861}a^{14}+\frac{29537}{2861}a^{12}-\frac{191}{2861}a^{10}-\frac{16965}{2861}a^{8}-\frac{45869}{2861}a^{6}-\frac{110304}{2861}a^{4}-\frac{9447}{2861}a^{2}+\frac{14626}{2861}$, $\frac{384}{2861}a^{19}+\frac{514}{2861}a^{17}+\frac{1177}{2861}a^{15}+\frac{3608}{2861}a^{13}-\frac{765}{2861}a^{11}-\frac{588}{2861}a^{9}-\frac{4297}{2861}a^{7}-\frac{13722}{2861}a^{5}+\frac{1887}{2861}a^{3}-\frac{5380}{2861}a$, $\frac{4084}{2861}a^{19}+\frac{5407}{2861}a^{17}+\frac{14157}{2861}a^{15}+\frac{32889}{2861}a^{13}-\frac{1073}{2861}a^{11}-\frac{17340}{2861}a^{9}-\frac{60214}{2861}a^{7}-\frac{120369}{2861}a^{5}-\frac{14710}{2861}a^{3}+\frac{16333}{2861}a$, $\frac{3245}{5722}a^{19}+\frac{1871}{2861}a^{18}+\frac{3375}{5722}a^{17}+\frac{4681}{5722}a^{16}+\frac{4880}{2861}a^{15}+\frac{12617}{5722}a^{14}+\frac{23635}{5722}a^{13}+\frac{14659}{2861}a^{12}-\frac{4674}{2861}a^{11}-\frac{3655}{5722}a^{10}-\frac{6016}{2861}a^{9}-\frac{15207}{5722}a^{8}-\frac{20745}{2861}a^{7}-\frac{27590}{2861}a^{6}-\frac{85247}{5722}a^{5}-\frac{110383}{5722}a^{4}+\frac{10957}{2861}a^{3}-\frac{1691}{2861}a^{2}+\frac{8925}{5722}a+\frac{10217}{5722}$, $\frac{2900}{2861}a^{18}+\frac{4299}{2861}a^{16}+\frac{8859}{2861}a^{14}+\frac{25579}{2861}a^{12}-\frac{3721}{2861}a^{10}-\frac{15527}{2861}a^{8}-\frac{36713}{2861}a^{6}-\frac{96656}{2861}a^{4}+\frac{8797}{2861}a^{2}+\frac{8126}{2861}$, $\frac{13049}{5722}a^{19}-\frac{2049}{5722}a^{18}+\frac{7802}{2861}a^{17}-\frac{3145}{5722}a^{16}+\frac{22185}{2861}a^{15}-\frac{3509}{2861}a^{14}+\frac{99301}{5722}a^{13}-\frac{9179}{2861}a^{12}-\frac{5812}{2861}a^{11}-\frac{813}{5722}a^{10}-\frac{25145}{2861}a^{9}+\frac{9843}{5722}a^{8}-\frac{188659}{5722}a^{7}+\frac{13800}{2861}a^{6}-\frac{358161}{5722}a^{5}+\frac{69867}{5722}a^{4}-\frac{8361}{2861}a^{3}+\frac{4722}{2861}a^{2}+\frac{23938}{2861}a-\frac{2410}{2861}$, $\frac{15007}{5722}a^{19}+\frac{1488}{2861}a^{18}+\frac{10081}{2861}a^{17}+\frac{2553}{5722}a^{16}+\frac{25372}{2861}a^{15}+\frac{9837}{5722}a^{14}+\frac{61412}{2861}a^{13}+\frac{19379}{5722}a^{12}-\frac{6897}{5722}a^{11}-\frac{3322}{2861}a^{10}-\frac{64911}{5722}a^{9}-\frac{3709}{2861}a^{8}-\frac{211657}{5722}a^{7}-\frac{21300}{2861}a^{6}-\frac{449497}{5722}a^{5}-\frac{64861}{5722}a^{4}-\frac{15240}{2861}a^{3}+\frac{2663}{2861}a^{2}+\frac{51855}{5722}a+\frac{3471}{2861}$, $\frac{7043}{2861}a^{19}+\frac{875}{2861}a^{18}+\frac{8727}{2861}a^{17}+\frac{1657}{5722}a^{16}+\frac{46647}{5722}a^{15}+\frac{2451}{2861}a^{14}+\frac{110773}{5722}a^{13}+\frac{6016}{2861}a^{12}-\frac{7929}{2861}a^{11}-\frac{5945}{5722}a^{10}-\frac{28040}{2861}a^{9}-\frac{3575}{2861}a^{8}-\frac{196143}{5722}a^{7}-\frac{9918}{2861}a^{6}-\frac{405991}{5722}a^{5}-\frac{40273}{5722}a^{4}+\frac{1820}{2861}a^{3}+\frac{12757}{5722}a^{2}+\frac{44821}{5722}a+\frac{824}{2861}$, $\frac{3245}{5722}a^{19}+\frac{548}{2861}a^{18}+\frac{3375}{5722}a^{17}+\frac{871}{5722}a^{16}+\frac{4880}{2861}a^{15}+\frac{4015}{5722}a^{14}+\frac{23635}{5722}a^{13}+\frac{3480}{2861}a^{12}-\frac{4674}{2861}a^{11}-\frac{1647}{5722}a^{10}-\frac{6016}{2861}a^{9}-\frac{963}{5722}a^{8}-\frac{20745}{2861}a^{7}-\frac{9321}{2861}a^{6}-\frac{85247}{5722}a^{5}-\frac{23787}{5722}a^{4}+\frac{10957}{2861}a^{3}-\frac{1688}{2861}a^{2}+\frac{8925}{5722}a-\frac{3673}{5722}$, $\frac{3409}{2861}a^{19}+\frac{65}{2861}a^{18}+\frac{6593}{5722}a^{17}+\frac{25}{5722}a^{16}+\frac{21181}{5722}a^{15}+\frac{460}{2861}a^{14}+\frac{23507}{2861}a^{13}+\frac{387}{5722}a^{12}-\frac{18813}{5722}a^{11}+\frac{474}{2861}a^{10}-\frac{23851}{5722}a^{9}+\frac{695}{5722}a^{8}-\frac{46514}{2861}a^{7}-\frac{2061}{2861}a^{6}-\frac{166837}{5722}a^{5}-\frac{801}{5722}a^{4}+\frac{18339}{2861}a^{3}-\frac{4055}{5722}a^{2}+\frac{19215}{5722}a+\frac{3245}{5722}$ (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: | \( 59115.4547078 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{7}\cdot 59115.4547078 \cdot 1}{2\cdot\sqrt{13167225195265942636954624}}\cr\approx \mathstrut & 0.201540791436 \end{aligned}\] (assuming GRH)
Galois group
$C_2^5.C_2^8:C_{10}$ (as 20T749):
A solvable group of order 81920 |
The 332 conjugacy class representatives for $C_2^5.C_2^8:C_{10}$ |
Character table for $C_2^5.C_2^8:C_{10}$ |
Intermediate fields
\(\Q(\zeta_{11})^+\), 10.6.113395848049.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.10.13 | $x^{10} + 10 x^{9} + 10 x^{8} + 56 x^{7} + 192 x^{6} + 800 x^{5} + 1536 x^{4} + 2208 x^{3} + 2224 x^{2} + 96 x - 1056$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |