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Properties

Label 44.0.653...000.3
Degree $44$
Signature $[0, 22]$
Discriminant $6.535\times 10^{85}$
Root discriminant \(89.20\)
Ramified primes $2,5,23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625)
 
gp: K = bnfinit(y^44 - 5*y^42 + 25*y^40 - 125*y^38 + 625*y^36 - 3125*y^34 + 15625*y^32 - 78125*y^30 + 390625*y^28 - 1953125*y^26 + 9765625*y^24 - 48828125*y^22 + 244140625*y^20 - 1220703125*y^18 + 6103515625*y^16 - 30517578125*y^14 + 152587890625*y^12 - 762939453125*y^10 + 3814697265625*y^8 - 19073486328125*y^6 + 95367431640625*y^4 - 476837158203125*y^2 + 2384185791015625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625)
 

\( x^{44} - 5 x^{42} + 25 x^{40} - 125 x^{38} + 625 x^{36} - 3125 x^{34} + 15625 x^{32} + \cdots + 23\!\cdots\!25 \) 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:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(653\!\cdots\!000\) \(\medspace = 2^{44}\cdot 5^{22}\cdot 23^{42}\) 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:  \(89.20\)
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:  $2\cdot 5^{1/2}23^{21/22}\approx 89.19615099241642$
Ramified primes:   \(2\), \(5\), \(23\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(460=2^{2}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(259,·)$, $\chi_{460}(261,·)$, $\chi_{460}(139,·)$, $\chi_{460}(141,·)$, $\chi_{460}(399,·)$, $\chi_{460}(401,·)$, $\chi_{460}(19,·)$, $\chi_{460}(21,·)$, $\chi_{460}(279,·)$, $\chi_{460}(281,·)$, $\chi_{460}(159,·)$, $\chi_{460}(419,·)$, $\chi_{460}(421,·)$, $\chi_{460}(39,·)$, $\chi_{460}(41,·)$, $\chi_{460}(301,·)$, $\chi_{460}(179,·)$, $\chi_{460}(181,·)$, $\chi_{460}(439,·)$, $\chi_{460}(441,·)$, $\chi_{460}(59,·)$, $\chi_{460}(61,·)$, $\chi_{460}(319,·)$, $\chi_{460}(321,·)$, $\chi_{460}(199,·)$, $\chi_{460}(201,·)$, $\chi_{460}(459,·)$, $\chi_{460}(79,·)$, $\chi_{460}(81,·)$, $\chi_{460}(339,·)$, $\chi_{460}(341,·)$, $\chi_{460}(219,·)$, $\chi_{460}(221,·)$, $\chi_{460}(99,·)$, $\chi_{460}(101,·)$, $\chi_{460}(359,·)$, $\chi_{460}(361,·)$, $\chi_{460}(239,·)$, $\chi_{460}(241,·)$, $\chi_{460}(119,·)$, $\chi_{460}(121,·)$, $\chi_{460}(379,·)$, $\chi_{460}(381,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{25}a^{4}$, $\frac{1}{25}a^{5}$, $\frac{1}{125}a^{6}$, $\frac{1}{125}a^{7}$, $\frac{1}{625}a^{8}$, $\frac{1}{625}a^{9}$, $\frac{1}{3125}a^{10}$, $\frac{1}{3125}a^{11}$, $\frac{1}{15625}a^{12}$, $\frac{1}{15625}a^{13}$, $\frac{1}{78125}a^{14}$, $\frac{1}{78125}a^{15}$, $\frac{1}{390625}a^{16}$, $\frac{1}{390625}a^{17}$, $\frac{1}{1953125}a^{18}$, $\frac{1}{1953125}a^{19}$, $\frac{1}{9765625}a^{20}$, $\frac{1}{9765625}a^{21}$, $\frac{1}{48828125}a^{22}$, $\frac{1}{48828125}a^{23}$, $\frac{1}{244140625}a^{24}$, $\frac{1}{244140625}a^{25}$, $\frac{1}{1220703125}a^{26}$, $\frac{1}{1220703125}a^{27}$, $\frac{1}{6103515625}a^{28}$, $\frac{1}{6103515625}a^{29}$, $\frac{1}{30517578125}a^{30}$, $\frac{1}{30517578125}a^{31}$, $\frac{1}{152587890625}a^{32}$, $\frac{1}{152587890625}a^{33}$, $\frac{1}{762939453125}a^{34}$, $\frac{1}{762939453125}a^{35}$, $\frac{1}{3814697265625}a^{36}$, $\frac{1}{3814697265625}a^{37}$, $\frac{1}{19073486328125}a^{38}$, $\frac{1}{19073486328125}a^{39}$, $\frac{1}{95367431640625}a^{40}$, $\frac{1}{95367431640625}a^{41}$, $\frac{1}{476837158203125}a^{42}$, $\frac{1}{476837158203125}a^{43}$ 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:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1}{3125} a^{10} \)  (order $46$) 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 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625)
 
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 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625, 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 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625);
 
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 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625);
 
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

$C_2\times C_{22}$ (as 44T2):

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)
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{115}) \), \(\Q(\sqrt{-5}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.0.351468714257323283030813737164800000000000.1, \(\Q(\zeta_{23})\), 22.22.8083780427918435509708715954790400000000000.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.11.0.1}{11} }^{4}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R ${\href{/padicField/29.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ ${\href{/padicField/41.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/padicField/47.1.0.1}{1} }^{44}$ $22^{2}$ $22^{2}$

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 Deg $22$$2$$11$$22$
Deg $22$$2$$11$$22$
\(5\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(23\) Copy content Toggle raw display 23.22.21.17$x^{22} + 23$$22$$1$$21$22T1$[\ ]_{22}$
23.22.21.17$x^{22} + 23$$22$$1$$21$22T1$[\ ]_{22}$