Normalized defining polynomial
\( x^{19} - x^{18} - 6 x^{17} + 7 x^{16} + 13 x^{15} - 21 x^{14} - 16 x^{13} + 41 x^{12} + 10 x^{11} + \cdots - 1 \)
Invariants
Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-23920741637672928000872108534200\)
\(\medspace = -\,2^{3}\cdot 5^{2}\cdot 430709\cdot 19005923\cdot 14610723851226353\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.82\) | 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))
| |
Galois root discriminant: | not computed | ||
Ramified primes: |
\(2\), \(5\), \(430709\), \(19005923\), \(14610723851226353\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-23920\!\cdots\!85342}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: |
$a$, $a^{18}-6a^{16}+a^{15}+14a^{14}-7a^{13}-23a^{12}+18a^{11}+28a^{10}-23a^{9}-18a^{8}+25a^{7}+6a^{6}-18a^{5}+5a^{3}-6a^{2}-4a$, $2a^{17}-a^{16}-12a^{15}+7a^{14}+27a^{13}-22a^{12}-40a^{11}+45a^{10}+45a^{9}-51a^{8}-31a^{7}+40a^{6}+9a^{5}-23a^{4}-4a^{3}+2a^{2}-3a+1$, $a^{18}+2a^{17}-9a^{16}-10a^{15}+32a^{14}+12a^{13}-63a^{12}+a^{11}+84a^{10}-6a^{9}-70a^{8}-2a^{7}+38a^{6}+10a^{5}-22a^{4}-19a^{3}+7a^{2}+7a-2$, $a^{18}-3a^{16}+a^{15}-a^{14}-2a^{13}+4a^{12}-7a^{11}-10a^{10}+19a^{9}+10a^{8}-18a^{7}+4a^{6}+18a^{5}-6a^{4}-2a^{3}+6a^{2}-a-2$, $a^{18}-a^{17}-5a^{16}+5a^{15}+9a^{14}-10a^{13}-15a^{12}+20a^{11}+20a^{10}-26a^{9}-17a^{8}+28a^{7}+5a^{6}-25a^{5}+a^{4}+12a^{3}-8a^{2}-2a+3$, $2a^{18}-4a^{17}-8a^{16}+23a^{15}+2a^{14}-49a^{13}+22a^{12}+67a^{11}-55a^{10}-51a^{9}+70a^{8}+17a^{7}-60a^{6}+10a^{5}+27a^{4}-16a^{3}-8a^{2}+10a-2$, $2a^{18}-14a^{16}+a^{15}+39a^{14}-10a^{13}-67a^{12}+36a^{11}+84a^{10}-56a^{9}-66a^{8}+51a^{7}+15a^{6}-40a^{5}+13a^{4}+23a^{3}-10a^{2}-6a+2$, $10a^{18}-7a^{17}-61a^{16}+51a^{15}+138a^{14}-162a^{13}-191a^{12}+328a^{11}+177a^{10}-407a^{9}-61a^{8}+351a^{7}-73a^{6}-217a^{5}+89a^{4}+59a^{3}-73a^{2}-5a+23$, $5a^{18}-34a^{16}+5a^{15}+91a^{14}-37a^{13}-154a^{12}+105a^{11}+198a^{10}-151a^{9}-159a^{8}+150a^{7}+64a^{6}-119a^{5}-13a^{4}+48a^{3}-15a^{2}-12a+5$, $7a^{18}-3a^{17}-45a^{16}+25a^{15}+112a^{14}-94a^{13}-176a^{12}+215a^{11}+198a^{10}-292a^{9}-123a^{8}+280a^{7}+a^{6}-197a^{5}+45a^{4}+69a^{3}-56a^{2}-12a+21$
| 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: | \( 840306267.085 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{7}\cdot 840306267.085 \cdot 1}{2\cdot\sqrt{23920741637672928000872108534200}}\cr\approx \mathstrut & 1.06274566845 \end{aligned}\] (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 | $15{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.9.0.1}{9} }$ | $15{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.15.0.1 | $x^{15} + x^{5} + x^{4} + x^{2} + 1$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(430709\)
| $\Q_{430709}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(19005923\)
| $\Q_{19005923}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(14610723851226353\)
| 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |