Normalized defining polynomial
\( x^{19} - 3 x^{18} - 12 x^{17} + 40 x^{16} + 52 x^{15} - 216 x^{14} - 88 x^{13} + 614 x^{12} + 8 x^{11} + \cdots + 1 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[13, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-6102326046691495371062145790782484\) \(\medspace = -\,2^{2}\cdot 90313\cdot 16881409121807\cdot 1000636759045331\) | 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: | \(60.00\) | 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: | $2\cdot 90313^{1/2}16881409121807^{1/2}1000636759045331^{1/2}\approx 7.811738632783035e+16$ | ||
Ramified primes: | \(2\), \(90313\), \(16881409121807\), \(1000636759045331\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-15255\!\cdots\!95621}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $15$ | 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: | $a$, $2a^{18}-6a^{17}-23a^{16}+77a^{15}+93a^{14}-395a^{13}-135a^{12}+1049a^{11}-31a^{10}-1539a^{9}+247a^{8}+1256a^{7}-222a^{6}-570a^{5}+85a^{4}+147a^{3}-15a^{2}-16a+1$, $43a^{18}-93a^{17}-617a^{16}+1274a^{15}+3571a^{14}-7219a^{13}-10920a^{12}+22077a^{11}+20371a^{10}-38456a^{9}-25286a^{8}+37258a^{7}+20288a^{6}-18790a^{5}-9257a^{4}+4425a^{3}+2059a^{2}-378a-169$, $a^{18}-2a^{17}-14a^{16}+26a^{15}+78a^{14}-138a^{13}-226a^{12}+388a^{11}+396a^{10}-591a^{9}-448a^{8}+456a^{7}+293a^{6}-168a^{5}-96a^{4}+32a^{3}+18a^{2}$, $21a^{18}-50a^{17}-283a^{16}+665a^{15}+1504a^{14}-3607a^{13}-4081a^{12}+10376a^{11}+6587a^{10}-16666a^{9}-7302a^{8}+14482a^{7}+5529a^{6}-6270a^{5}-2363a^{4}+1229a^{3}+466a^{2}-91a-35$, $9a^{18}-23a^{17}-117a^{16}+305a^{15}+587a^{14}-1642a^{13}-1435a^{12}+4659a^{11}+1920a^{10}-7356a^{9}-1670a^{8}+6291a^{7}+1051a^{6}-2693a^{5}-371a^{4}+524a^{3}+40a^{2}-39a+1$, $209a^{18}-759a^{17}-2056a^{16}+9741a^{15}+5036a^{14}-49402a^{13}+11483a^{12}+126720a^{11}-76365a^{10}-173402a^{9}+139869a^{8}+123583a^{7}-116120a^{6}-41681a^{5}+44915a^{4}+6058a^{3}-7827a^{2}-286a+495$, $283a^{18}-691a^{17}-3783a^{16}+9212a^{15}+19871a^{14}-50082a^{13}-52901a^{12}+144447a^{11}+82927a^{10}-233527a^{9}-89857a^{8}+206314a^{7}+69077a^{6}-92443a^{5}-31372a^{4}+19024a^{3}+6727a^{2}-1428a-523$, $44a^{18}-222a^{17}-238a^{16}+2752a^{15}-1460a^{14}-13081a^{13}+15422a^{12}+29576a^{11}-50590a^{10}-31448a^{9}+79357a^{8}+11894a^{7}-63103a^{6}+2337a^{5}+24427a^{4}-1889a^{3}-4421a^{2}+234a+300$, $114a^{18}-263a^{17}-1570a^{16}+3531a^{15}+8607a^{14}-19430a^{13}-24465a^{12}+57077a^{11}+41958a^{10}-94384a^{9}-48908a^{8}+85454a^{7}+37979a^{6}-39311a^{5}-16682a^{4}+8295a^{3}+3458a^{2}-640a-263$, $138a^{18}-428a^{17}-1580a^{16}+5583a^{15}+6226a^{14}-29167a^{13}-7577a^{12}+78816a^{11}-9405a^{10}-117018a^{9}+31359a^{8}+94149a^{7}-27593a^{6}-38424a^{5}+9685a^{4}+7388a^{3}-1337a^{2}-530a+50$, $77a^{18}-309a^{17}-670a^{16}+3935a^{15}+712a^{14}-19657a^{13}+10231a^{12}+48997a^{11}-44486a^{10}-63824a^{9}+76094a^{8}+41835a^{7}-62812a^{6}-11944a^{5}+24850a^{4}+1181a^{3}-4510a^{2}-5a+300$, $18a^{18}-65a^{17}-181a^{16}+846a^{15}+473a^{14}-4382a^{13}+882a^{12}+11614a^{11}-6708a^{10}-16764a^{9}+13221a^{8}+13113a^{7}-12196a^{6}-5288a^{5}+5594a^{4}+1046a^{3}-1228a^{2}-80a+102$, $26a^{18}-42a^{17}-414a^{16}+597a^{15}+2700a^{14}-3590a^{13}-9487a^{12}+11892a^{11}+20364a^{10}-22470a^{9}-27772a^{8}+23312a^{7}+22877a^{6}-12406a^{5}-10218a^{4}+3041a^{3}+2183a^{2}-270a-172$, $158a^{18}-544a^{17}-1655a^{16}+7053a^{15}+5092a^{14}-36381a^{13}+2206a^{12}+96020a^{11}-41237a^{10}-137641a^{9}+83471a^{8}+105803a^{7}-72482a^{6}-40686a^{5}+29294a^{4}+7230a^{3}-5375a^{2}-459a+356$ (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: | \( 61782008553.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{3}\cdot 61782008553.5 \cdot 1}{2\cdot\sqrt{6102326046691495371062145790782484}}\cr\approx \mathstrut & 0.803551788717 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 121645100408832000 |
The 490 conjugacy class representatives for $S_{19}$ |
Character table for $S_{19}$ |
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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(90313\) | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(16881409121807\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(1000636759045331\) | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |