Normalized defining polynomial
\( x^{19} - 3 x^{18} - x^{17} + 15 x^{16} - 22 x^{15} - 6 x^{14} + 59 x^{13} - 70 x^{12} - 6 x^{11} + \cdots - 1 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[7, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(8322004363062341150231840635904\) \(\medspace = 2^{10}\cdot 2746819\cdot 2958679616604904629509\) | 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: | \(42.40\) | 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: | $2^{3/2}2746819^{1/2}2958679616604904629509^{1/2}\approx 254981683825377.03$ | ||
Ramified primes: | \(2\), \(2746819\), \(2958679616604904629509\) | 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: | $\Q(\sqrt{81269\!\cdots\!81871}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$
Monogenic: | Yes | |
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: | $a$, $6a^{18}-16a^{17}-11a^{16}+85a^{15}-103a^{14}-65a^{13}+320a^{12}-308a^{11}-117a^{10}+554a^{9}-436a^{8}-114a^{7}+443a^{6}-249a^{5}-58a^{4}+137a^{3}-46a^{2}-11a+6$, $a^{18}-2a^{17}-3a^{16}+12a^{15}-10a^{14}-16a^{13}+43a^{12}-27a^{11}-33a^{10}+73a^{9}-34a^{8}-35a^{7}+57a^{6}-15a^{5}-16a^{4}+19a^{3}-a+2$, $9a^{18}-22a^{17}-22a^{16}+125a^{15}-128a^{14}-135a^{13}+472a^{12}-369a^{11}-292a^{10}+843a^{9}-514a^{8}-340a^{7}+712a^{6}-287a^{5}-191a^{4}+246a^{3}-51a^{2}-40a+19$, $a^{18}-2a^{17}-4a^{16}+13a^{15}-4a^{14}-26a^{13}+37a^{12}+5a^{11}-60a^{10}+46a^{9}+38a^{8}-72a^{7}+7a^{6}+58a^{5}-28a^{4}-19a^{3}+17a^{2}-4$, $8a^{18}-22a^{17}-12a^{16}+112a^{15}-147a^{14}-65a^{13}+416a^{12}-442a^{11}-92a^{10}+702a^{9}-619a^{8}-74a^{7}+551a^{6}-351a^{5}-46a^{4}+176a^{3}-66a^{2}-13a+11$, $3a^{18}-11a^{17}+4a^{16}+43a^{15}-93a^{14}+38a^{13}+154a^{12}-297a^{11}+153a^{10}+218a^{9}-419a^{8}+217a^{7}+131a^{6}-236a^{5}+94a^{4}+35a^{3}-41a^{2}+6a+2$, $7a^{18}-19a^{17}-12a^{16}+100a^{15}-125a^{14}-71a^{13}+379a^{12}-378a^{11}-123a^{10}+660a^{9}-543a^{8}-115a^{7}+535a^{6}-321a^{5}-59a^{4}+172a^{3}-64a^{2}-13a+9$, $8a^{18}-20a^{17}-20a^{16}+117a^{15}-120a^{14}-133a^{13}+460a^{12}-358a^{11}-300a^{10}+855a^{9}-533a^{8}-341a^{7}+743a^{6}-326a^{5}-179a^{4}+251a^{3}-59a^{2}-39a+19$, $17a^{18}-43a^{17}-36a^{16}+234a^{15}-264a^{14}-208a^{13}+874a^{12}-777a^{11}-401a^{10}+1516a^{9}-1082a^{8}-425a^{7}+1227a^{6}-602a^{5}-240a^{4}+410a^{3}-105a^{2}-60a+32$, $3a^{18}-5a^{17}-9a^{16}+31a^{15}-28a^{14}-42a^{13}+116a^{12}-80a^{11}-75a^{10}+206a^{9}-115a^{8}-67a^{7}+172a^{6}-75a^{5}-32a^{4}+62a^{3}-21a^{2}-9a+7$, $4a^{18}-11a^{17}-5a^{16}+51a^{15}-71a^{14}-12a^{13}+168a^{12}-210a^{11}+27a^{10}+228a^{9}-266a^{8}+69a^{7}+111a^{6}-118a^{5}+35a^{4}+13a^{3}-10a^{2}+2a$ (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: | \( 409832342.529 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{6}\cdot 409832342.529 \cdot 1}{2\cdot\sqrt{8322004363062341150231840635904}}\cr\approx \mathstrut & 0.559437330606 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 121645100408832000 |
The 490 conjugacy class representatives for $S_{19}$ are not computed |
Character table for $S_{19}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | $17{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $19$ | $16{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
2.4.6.9 | $x^{4} + 2 x^{3} + 6$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ | |
2.11.0.1 | $x^{11} + x^{2} + 1$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(2746819\) | 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 $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(295\!\cdots\!509\) | $\Q_{29\!\cdots\!09}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |