Properties

Label 44.2.314...432.1
Degree $44$
Signature $[2, 21]$
Discriminant $-3.140\times 10^{84}$
Root discriminant \(83.25\)
Ramified primes $2,3,11$
Class number not computed
Class group not computed
Galois group $D_4\times F_{11}$ (as 44T32)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 3)
 
gp: K = bnfinit(y^44 - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 3);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 3)
 

\( x^{44} - 3 \) Copy content Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-314\!\cdots\!432\) \(\medspace = -\,2^{88}\cdot 3^{43}\cdot 11^{36}\) 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:  \(83.25\)
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:  $2^{2}3^{43/44}11^{9/10}\approx 101.29569716579633$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

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

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

Monogenic:  Not computed
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:  $22$
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^44 - 3)
 
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^44 - 3, 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^44 - 3);
 
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^44 - 3);
 
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

$D_4\times F_{11}$ (as 44T32):

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 solvable group of order 880
The 55 conjugacy class representatives for $D_4\times F_{11}$ are not computed
Character table for $D_4\times F_{11}$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.2.6912.1, 11.1.139234453205859.1, 22.2.243935263341322672277116036916969472.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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]
 

Sibling fields

Degree 44 siblings: deg 44, 44.0.748738788633404023046417283703695162922995794376200461450129397733922342699008.1, deg 44
Minimal sibling: 44.0.748738788633404023046417283703695162922995794376200461450129397733922342699008.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R $20^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.10.0.1}{10} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ R ${\href{/padicField/13.10.0.1}{10} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ $20^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.10.0.1}{10} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ $22{,}\,{\href{/padicField/23.11.0.1}{11} }^{2}$ $20^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.10.0.1}{10} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ $20^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.2.0.1}{2} }^{22}$ ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $20^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(2\) Copy content Toggle raw display 2.4.8.5$x^{4} + 2 x^{2} + 4 x + 6$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
Deg $40$$4$$10$$80$
\(3\) Copy content Toggle raw display Deg $44$$44$$1$$43$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$