Normalized defining polynomial
\( x^{21} + 21 x^{19} + 189 x^{17} + 952 x^{15} + 2940 x^{13} + 5733 x^{11} + 7007 x^{9} + 5144 x^{7} + \cdots - 1 \)
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: | \(39987268032456435383175526376221\) \(\medspace = 7^{19}\cdot 23^{7}\cdot 101^{3}\) | 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: | \(31.98\) | 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: | $7^{341/294}23^{1/2}101^{1/2}\approx 460.4933384155175$ | ||
Ramified primes: | \(7\), \(23\), \(101\) | 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: | \(\Q(\sqrt{16261}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{7}a^{20}-\frac{2}{7}a^{19}-\frac{3}{7}a^{18}-\frac{1}{7}a^{17}+\frac{2}{7}a^{16}+\frac{3}{7}a^{15}+\frac{1}{7}a^{14}-\frac{2}{7}a^{13}-\frac{3}{7}a^{12}-\frac{1}{7}a^{11}+\frac{2}{7}a^{10}+\frac{3}{7}a^{9}+\frac{1}{7}a^{8}-\frac{2}{7}a^{7}+\frac{3}{7}a^{6}+\frac{1}{7}a^{5}-\frac{2}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}+\frac{2}{7}a+\frac{3}{7}$
Monogenic: | Not computed | |
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: | $a^{14}+14a^{12}+77a^{10}+210a^{8}+294a^{6}+196a^{4}+49a^{2}-1$, $a$, $\frac{1}{7}a^{20}-\frac{2}{7}a^{19}+\frac{25}{7}a^{18}-\frac{36}{7}a^{17}+\frac{261}{7}a^{16}-\frac{270}{7}a^{15}+\frac{1492}{7}a^{14}-\frac{1094}{7}a^{13}+\frac{5128}{7}a^{12}-\frac{2605}{7}a^{11}+\frac{10929}{7}a^{10}-\frac{3749}{7}a^{9}+\frac{14351}{7}a^{8}-\frac{3306}{7}a^{7}+\frac{11140}{7}a^{6}-\frac{1840}{7}a^{5}+\frac{4646}{7}a^{4}-\frac{640}{7}a^{3}+\frac{797}{7}a^{2}-\frac{110}{7}a+\frac{3}{7}$, $\frac{5}{7}a^{20}-\frac{3}{7}a^{19}+\frac{90}{7}a^{18}-\frac{54}{7}a^{17}+\frac{668}{7}a^{16}-\frac{405}{7}a^{15}+\frac{2630}{7}a^{14}-\frac{1641}{7}a^{13}+\frac{5872}{7}a^{12}-\frac{3897}{7}a^{11}+\frac{7325}{7}a^{10}-\frac{5508}{7}a^{9}+\frac{4583}{7}a^{8}-\frac{4497}{7}a^{7}+\frac{897}{7}a^{6}-\frac{1948}{7}a^{5}-\frac{325}{7}a^{4}-\frac{358}{7}a^{3}-\frac{117}{7}a^{2}-\frac{11}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{20}-\frac{2}{7}a^{19}+\frac{18}{7}a^{18}-\frac{36}{7}a^{17}+\frac{135}{7}a^{16}-\frac{270}{7}a^{15}+\frac{540}{7}a^{14}-\frac{1087}{7}a^{13}+\frac{1201}{7}a^{12}-\frac{2507}{7}a^{11}+\frac{1290}{7}a^{10}-\frac{3217}{7}a^{9}-\frac{41}{7}a^{8}-\frac{1906}{7}a^{7}-\frac{1649}{7}a^{6}-\frac{27}{7}a^{5}-\frac{1570}{7}a^{4}+\frac{389}{7}a^{3}-\frac{456}{7}a^{2}+\frac{86}{7}a-\frac{4}{7}$, $\frac{1}{7}a^{20}+\frac{5}{7}a^{19}+\frac{18}{7}a^{18}+\frac{90}{7}a^{17}+\frac{135}{7}a^{16}+\frac{675}{7}a^{15}+\frac{547}{7}a^{14}+\frac{2728}{7}a^{13}+\frac{1299}{7}a^{12}+\frac{6411}{7}a^{11}+\frac{1836}{7}a^{10}+\frac{8795}{7}a^{9}+\frac{1499}{7}a^{8}+\frac{6641}{7}a^{7}+\frac{647}{7}a^{6}+\frac{2269}{7}a^{5}+\frac{103}{7}a^{4}+\frac{18}{7}a^{3}-\frac{29}{7}a^{2}-\frac{124}{7}a-\frac{18}{7}$, $\frac{4}{7}a^{20}-\frac{8}{7}a^{19}+\frac{79}{7}a^{18}-\frac{144}{7}a^{17}+\frac{659}{7}a^{16}-\frac{1080}{7}a^{15}+\frac{3028}{7}a^{14}-\frac{4369}{7}a^{13}+\frac{8395}{7}a^{12}-\frac{10315}{7}a^{11}+\frac{14505}{7}a^{10}-\frac{14373}{7}a^{9}+\frac{15628}{7}a^{8}-\frac{11390}{7}a^{7}+\frac{10197}{7}a^{6}-\frac{4609}{7}a^{5}+\frac{3681}{7}a^{4}-\frac{621}{7}a^{3}+\frac{570}{7}a^{2}+\frac{64}{7}a+\frac{12}{7}$, $\frac{4}{7}a^{20}+\frac{6}{7}a^{19}+\frac{72}{7}a^{18}+\frac{108}{7}a^{17}+\frac{540}{7}a^{16}+\frac{817}{7}a^{15}+\frac{2181}{7}a^{14}+\frac{3373}{7}a^{13}+\frac{5112}{7}a^{12}+\frac{8256}{7}a^{11}+\frac{6959}{7}a^{10}+\frac{12185}{7}a^{9}+\frac{5142}{7}a^{8}+\frac{10562}{7}a^{7}+\frac{1622}{7}a^{6}+\frac{4981}{7}a^{5}-\frac{92}{7}a^{4}+\frac{1031}{7}a^{3}-\frac{123}{7}a^{2}+\frac{22}{7}a-\frac{2}{7}$, $\frac{2}{7}a^{20}-\frac{4}{7}a^{19}+\frac{36}{7}a^{18}-\frac{86}{7}a^{17}+\frac{263}{7}a^{16}-\frac{778}{7}a^{15}+\frac{975}{7}a^{14}-\frac{3847}{7}a^{13}+\frac{1779}{7}a^{12}-\frac{11286}{7}a^{11}+\frac{753}{7}a^{10}-\frac{19902}{7}a^{9}-\frac{2714}{7}a^{8}-\frac{20297}{7}a^{7}-\frac{4677}{7}a^{6}-\frac{10736}{7}a^{5}-\frac{2741}{7}a^{4}-\frac{2204}{7}a^{3}-\frac{513}{7}a^{2}+\frac{46}{7}a+\frac{41}{7}$, $\frac{10}{7}a^{20}+\frac{1}{7}a^{19}+\frac{208}{7}a^{18}+\frac{32}{7}a^{17}+\frac{1854}{7}a^{16}+\frac{366}{7}a^{15}+\frac{9236}{7}a^{14}+\frac{2122}{7}a^{13}+\frac{28117}{7}a^{12}+\frac{7032}{7}a^{11}+\frac{53731}{7}a^{10}+\frac{13848}{7}a^{9}+\frac{63780}{7}a^{8}+\frac{16073}{7}a^{7}+\frac{44977}{7}a^{6}+\frac{10496}{7}a^{5}+\frac{17158}{7}a^{4}+\frac{3533}{7}a^{3}+\frac{2769}{7}a^{2}+\frac{475}{7}a+\frac{37}{7}$ (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: | \( 28539748.2764 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 28539748.2764 \cdot 1}{2\cdot\sqrt{39987268032456435383175526376221}}\cr\approx \mathstrut & 0.432800700466 \end{aligned}\] (assuming GRH)
Galois group
$C_7^3:(C_6\times S_4)$ (as 21T87):
A solvable group of order 49392 |
The 51 conjugacy class representatives for $C_7^3:(C_6\times S_4)$ |
Character table for $C_7^3:(C_6\times S_4)$ |
Intermediate fields
3.1.23.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 28 sibling: | data not computed |
Degree 42 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 | ${\href{/padicField/2.6.0.1}{6} }^{3}{,}\,{\href{/padicField/2.3.0.1}{3} }$ | $21$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $21$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.14.14.13 | $x^{14} + 28 x^{9} - 154 x^{8} + 14 x^{7} + 196 x^{4} - 2156 x^{3} - 1225 x^{2} - 1078 x + 49$ | $7$ | $2$ | $14$ | 14T23 | $[7/6, 7/6]_{6}^{2}$ | |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(101\) | $\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
101.6.3.2 | $x^{6} + 30603 x^{2} - 101999799$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
101.6.0.1 | $x^{6} + 2 x^{4} + 90 x^{3} + 20 x^{2} + 67 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
101.6.0.1 | $x^{6} + 2 x^{4} + 90 x^{3} + 20 x^{2} + 67 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |