Normalized defining polynomial
\( x^{19} - 3 x^{18} - x^{17} + 15 x^{16} - 22 x^{15} - 6 x^{14} + 59 x^{13} - 70 x^{12} - 6 x^{11} + 106 x^{10} - 107 x^{9} + 92 x^{7} - 73 x^{6} - x^{5} + 35 x^{4} - 19 x^{3} - x^{2} + 4 x - 1 \)
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)
| |
Discriminant: | \(105491601716452677126377141804\) \(\medspace = 2^{2}\cdot 131\cdot 149\cdot 397\cdot 20117\cdot 218819\cdot 773146117559\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.69\) | 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: | $2\cdot 131^{1/2}149^{1/2}397^{1/2}20117^{1/2}218819^{1/2}773146117559^{1/2}\approx 324794707032692.7$ | ||
Ramified primes: | \(2\), \(131\), \(149\), \(397\), \(20117\), \(218819\), \(773146117559\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{26372\!\cdots\!85451}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $12$ | 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: | $a$, $a^{18}-3a^{17}-a^{16}+14a^{15}-20a^{14}-3a^{13}+48a^{12}-61a^{11}+6a^{10}+70a^{9}-82a^{8}+19a^{7}+41a^{6}-42a^{5}+11a^{4}+6a^{3}-6a^{2}+a$, $a^{17}-2a^{16}-4a^{15}+13a^{14}-5a^{13}-24a^{12}+40a^{11}-6a^{10}-52a^{9}+60a^{8}+5a^{7}-55a^{6}+32a^{5}+14a^{4}-19a^{3}+2a^{2}+2a-1$, $2a^{18}-5a^{17}-5a^{16}+29a^{15}-29a^{14}-34a^{13}+112a^{12}-81a^{11}-82a^{10}+206a^{9}-108a^{8}-107a^{7}+184a^{6}-54a^{5}-74a^{4}+68a^{3}-5a^{2}-18a+6$, $a^{18}-3a^{17}-a^{16}+14a^{15}-19a^{14}-5a^{13}+45a^{12}-51a^{11}-a^{10}+61a^{9}-56a^{8}+a^{7}+31a^{6}-17a^{5}-2a^{4}+4a^{3}-2a^{2}+a$, $2a^{18}-2a^{17}-10a^{16}+16a^{15}+5a^{14}-41a^{13}+42a^{12}+20a^{11}-78a^{10}+52a^{9}+40a^{8}-57a^{7}+17a^{6}+26a^{5}-10a^{4}-3a^{3}+4a^{2}+a-1$, $2a^{18}-21a^{16}+30a^{15}+37a^{14}-157a^{13}+143a^{12}+141a^{11}-460a^{10}+373a^{9}+182a^{8}-608a^{7}+442a^{6}+78a^{5}-320a^{4}+184a^{3}+8a^{2}-49a+14$, $13a^{18}-37a^{17}-18a^{16}+186a^{15}-245a^{14}-103a^{13}+676a^{12}-718a^{11}-140a^{10}+1100a^{9}-954a^{8}-108a^{7}+801a^{6}-476a^{5}-94a^{4}+214a^{3}-57a^{2}-22a+11$, $a^{18}-4a^{17}+20a^{15}-31a^{14}-7a^{13}+84a^{12}-101a^{11}-15a^{10}+164a^{9}-160a^{8}-18a^{7}+154a^{6}-115a^{5}-18a^{4}+61a^{3}-27a^{2}-7a+6$, $4a^{18}-9a^{17}-12a^{16}+55a^{15}-47a^{14}-76a^{13}+210a^{12}-129a^{11}-179a^{10}+385a^{9}-175a^{8}-214a^{7}+338a^{6}-95a^{5}-125a^{4}+120a^{3}-14a^{2}-26a+10$, $9a^{18}-21a^{17}-29a^{16}+134a^{15}-105a^{14}-207a^{13}+521a^{12}-275a^{11}-520a^{10}+983a^{9}-368a^{8}-648a^{7}+874a^{6}-191a^{5}-360a^{4}+288a^{3}-22a^{2}-54a+18$, $5a^{18}-8a^{17}-21a^{16}+54a^{15}-15a^{14}-107a^{13}+167a^{12}-11a^{11}-227a^{10}+246a^{9}+37a^{8}-230a^{7}+139a^{6}+66a^{5}-87a^{4}+11a^{3}+14a^{2}-6a$ (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: | \( 52861133.6694 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{6}\cdot 52861133.6694 \cdot 1}{2\cdot\sqrt{105491601716452677126377141804}}\cr\approx \mathstrut & 0.640894749664 \end{aligned}\] (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 | $17{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(131\) | 131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
131.7.0.1 | $x^{7} + 10 x + 129$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
131.10.0.1 | $x^{10} + 124 x^{5} + 97 x^{4} + 9 x^{3} + 126 x^{2} + 44 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(149\) | 149.2.1.1 | $x^{2} + 149$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
149.7.0.1 | $x^{7} + 19 x + 147$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
149.10.0.1 | $x^{10} + 74 x^{5} + 42 x^{4} + 148 x^{3} + 143 x^{2} + 51 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(397\) | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(20117\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(218819\) | $\Q_{218819}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{218819}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{218819}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{218819}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(773146117559\) | $\Q_{773146117559}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{773146117559}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |