Properties

Label 45.3.582...875.1
Degree $45$
Signature $[3, 21]$
Discriminant $-5.829\times 10^{103}$
Root discriminant \(202.26\)
Ramified primes see page
Class number not computed
Class group not computed
Galois group $S_{45}$ (as 45T10923)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 5*x - 1)
 
gp: K = bnfinit(y^45 - 5*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 5*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 5*x - 1)
 

\( x^{45} - 5x - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[3, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-582\!\cdots\!875\) \(\medspace = -\,5^{45}\cdot 13\cdot 47\cdot 241\cdot 313\cdot 10159\cdot 1560681193\cdot 441481146991\cdot 24561359840613046207\cdot 258812619101427272051\) 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:  \(202.26\)
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))
 
Ramified primes:   \(5\), \(13\), \(47\), \(241\), \(313\), \(10159\), \(1560681193\), \(441481146991\), \(24561359840613046207\), \(258812619101427272051\) 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:  $\Q(\sqrt{-10253\!\cdots\!29235}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$ 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

not computed

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:  $23$
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^45 - 5*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^45 - 5*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^45 - 5*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^45 - 5*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

$S_{45}$ (as 45T10923):

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 119622220865480194561963161495657715064383733760000000000
The 89134 conjugacy class representatives for $S_{45}$ are not computed
Character table for $S_{45}$ 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 $22{,}\,15{,}\,{\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ $36{,}\,{\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ R $27{,}\,18$ $16{,}\,{\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ R $29{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ $26{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ $23{,}\,{\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ $27{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $22{,}\,{\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.12.0.1}{12} }^{3}{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}$ $30{,}\,{\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ R $32{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$

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
\(5\) Copy content Toggle raw display 5.5.5.4$x^{5} + 10 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.10$x^{10} + 40 x^{6} + 10 x^{5} - 200 x^{4} + 400 x^{2} + 200 x + 25$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
Deg $30$$5$$6$$30$
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.0.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.6.0.1$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.10.0.1$x^{10} + 7 x^{5} + 5 x^{4} + 8 x^{3} + x^{2} + x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
13.22.0.1$x^{22} + 2 x^{2} - 4 x + 7$$1$$22$$0$22T1$[\ ]^{22}$
\(47\) Copy content Toggle raw display $\Q_{47}$$x + 42$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.1$x^{2} + 235$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
47.3.0.1$x^{3} + 3 x + 42$$1$$3$$0$$C_3$$[\ ]^{3}$
47.3.0.1$x^{3} + 3 x + 42$$1$$3$$0$$C_3$$[\ ]^{3}$
47.4.0.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
47.12.0.1$x^{12} + 46 x^{7} + 40 x^{6} + 35 x^{5} + 12 x^{4} + 46 x^{3} + 14 x^{2} + 9 x + 5$$1$$12$$0$$C_{12}$$[\ ]^{12}$
47.14.0.1$x^{14} + 36 x^{7} + 20 x^{6} + 30 x^{5} + 17 x^{4} + 24 x^{3} + 9 x^{2} + 32 x + 5$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(241\) Copy content Toggle raw display $\Q_{241}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $27$$1$$27$$0$$C_{27}$$[\ ]^{27}$
\(313\) Copy content Toggle raw display $\Q_{313}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $20$$1$$20$$0$20T1$[\ ]^{20}$
Deg $22$$1$$22$$0$22T1$[\ ]^{22}$
\(10159\) Copy content Toggle raw display $\Q_{10159}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{10159}$$x$$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$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $31$$1$$31$$0$$C_{31}$$[\ ]^{31}$
\(1560681193\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$
\(441481146991\) Copy content Toggle raw display $\Q_{441481146991}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $23$$1$$23$$0$$C_{23}$$[\ ]^{23}$
\(24561359840613046207\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $43$$1$$43$$0$$C_{43}$$[\ ]^{43}$
\(258\!\cdots\!051\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $37$$1$$37$$0$$C_{37}$$[\ ]^{37}$