Properties

Label 19.7.105...804.1
Degree $19$
Signature $[7, 6]$
Discriminant $1.055\times 10^{29}$
Root discriminant \(33.69\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{19}$ (as 19T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(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)
 
gp: K = bnfinit(y^19 - 3*y^18 - y^17 + 15*y^16 - 22*y^15 - 6*y^14 + 59*y^13 - 70*y^12 - 6*y^11 + 106*y^10 - 107*y^9 + 92*y^7 - 73*y^6 - y^5 + 35*y^4 - 19*y^3 - y^2 + 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(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);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(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)
 

\( x^{19} - 3 x^{18} - x^{17} + 15 x^{16} - 22 x^{15} - 6 x^{14} + 59 x^{13} - 70 x^{12} - 6 x^{11} + 106 x^{10} - 107 x^{9} + 92 x^{7} - 73 x^{6} - x^{5} + 35 x^{4} - 19 x^{3} - x^{2} + 4 x - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $19$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[7, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(105491601716452677126377141804\) \(\medspace = 2^{2}\cdot 131\cdot 149\cdot 397\cdot 20117\cdot 218819\cdot 773146117559\) Copy content Toggle raw display
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:   \(2\), \(131\), \(149\), \(397\), \(20117\), \(218819\), \(773146117559\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $a$, $a^{18}-3a^{17}-a^{16}+14a^{15}-20a^{14}-3a^{13}+48a^{12}-61a^{11}+6a^{10}+70a^{9}-82a^{8}+19a^{7}+41a^{6}-42a^{5}+11a^{4}+6a^{3}-6a^{2}+a$, $a^{17}-2a^{16}-4a^{15}+13a^{14}-5a^{13}-24a^{12}+40a^{11}-6a^{10}-52a^{9}+60a^{8}+5a^{7}-55a^{6}+32a^{5}+14a^{4}-19a^{3}+2a^{2}+2a-1$, $2a^{18}-5a^{17}-5a^{16}+29a^{15}-29a^{14}-34a^{13}+112a^{12}-81a^{11}-82a^{10}+206a^{9}-108a^{8}-107a^{7}+184a^{6}-54a^{5}-74a^{4}+68a^{3}-5a^{2}-18a+6$, $a^{18}-3a^{17}-a^{16}+14a^{15}-19a^{14}-5a^{13}+45a^{12}-51a^{11}-a^{10}+61a^{9}-56a^{8}+a^{7}+31a^{6}-17a^{5}-2a^{4}+4a^{3}-2a^{2}+a$, $2a^{18}-2a^{17}-10a^{16}+16a^{15}+5a^{14}-41a^{13}+42a^{12}+20a^{11}-78a^{10}+52a^{9}+40a^{8}-57a^{7}+17a^{6}+26a^{5}-10a^{4}-3a^{3}+4a^{2}+a-1$, $2a^{18}-21a^{16}+30a^{15}+37a^{14}-157a^{13}+143a^{12}+141a^{11}-460a^{10}+373a^{9}+182a^{8}-608a^{7}+442a^{6}+78a^{5}-320a^{4}+184a^{3}+8a^{2}-49a+14$, $13a^{18}-37a^{17}-18a^{16}+186a^{15}-245a^{14}-103a^{13}+676a^{12}-718a^{11}-140a^{10}+1100a^{9}-954a^{8}-108a^{7}+801a^{6}-476a^{5}-94a^{4}+214a^{3}-57a^{2}-22a+11$, $a^{18}-4a^{17}+20a^{15}-31a^{14}-7a^{13}+84a^{12}-101a^{11}-15a^{10}+164a^{9}-160a^{8}-18a^{7}+154a^{6}-115a^{5}-18a^{4}+61a^{3}-27a^{2}-7a+6$, $4a^{18}-9a^{17}-12a^{16}+55a^{15}-47a^{14}-76a^{13}+210a^{12}-129a^{11}-179a^{10}+385a^{9}-175a^{8}-214a^{7}+338a^{6}-95a^{5}-125a^{4}+120a^{3}-14a^{2}-26a+10$, $9a^{18}-21a^{17}-29a^{16}+134a^{15}-105a^{14}-207a^{13}+521a^{12}-275a^{11}-520a^{10}+983a^{9}-368a^{8}-648a^{7}+874a^{6}-191a^{5}-360a^{4}+288a^{3}-22a^{2}-54a+18$, $5a^{18}-8a^{17}-21a^{16}+54a^{15}-15a^{14}-107a^{13}+167a^{12}-11a^{11}-227a^{10}+246a^{9}+37a^{8}-230a^{7}+139a^{6}+66a^{5}-87a^{4}+11a^{3}+14a^{2}-6a$ Copy content Toggle raw display (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)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(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, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$S_{19}$ (as 19T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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 $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 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}$
\(131\) Copy content Toggle raw display 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}$
\(149\) Copy content Toggle raw display 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}$
\(397\) Copy content Toggle raw display 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}$
\(20117\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(218819\) Copy content Toggle raw display $\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}$
\(773146117559\) Copy content Toggle raw display $\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}$