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Properties

Label 40.0.675...625.1
Degree $40$
Signature $[0, 20]$
Discriminant $6.759\times 10^{65}$
Root discriminant \(44.23\)
Ramified primes $5,7$
Class number not computed
Class group not computed
Galois group $C_2\times C_{20}$ (as 40T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 11*x^35 + 89*x^30 - 627*x^25 + 4049*x^20 - 20064*x^15 + 91136*x^10 - 360448*x^5 + 1048576)
 
gp: K = bnfinit(y^40 - 11*y^35 + 89*y^30 - 627*y^25 + 4049*y^20 - 20064*y^15 + 91136*y^10 - 360448*y^5 + 1048576, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 11*x^35 + 89*x^30 - 627*x^25 + 4049*x^20 - 20064*x^15 + 91136*x^10 - 360448*x^5 + 1048576);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 11*x^35 + 89*x^30 - 627*x^25 + 4049*x^20 - 20064*x^15 + 91136*x^10 - 360448*x^5 + 1048576)
 

\( x^{40} - 11x^{35} + 89x^{30} - 627x^{25} + 4049x^{20} - 20064x^{15} + 91136x^{10} - 360448x^{5} + 1048576 \) Copy content Toggle raw display

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

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(675866784901662726441095609131171073613586486317217350006103515625\) \(\medspace = 5^{70}\cdot 7^{20}\) 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:  \(44.23\)
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:  $5^{7/4}7^{1/2}\approx 44.23301346632757$
Ramified primes:   \(5\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(175=5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{175}(1,·)$, $\chi_{175}(132,·)$, $\chi_{175}(134,·)$, $\chi_{175}(8,·)$, $\chi_{175}(139,·)$, $\chi_{175}(13,·)$, $\chi_{175}(146,·)$, $\chi_{175}(148,·)$, $\chi_{175}(22,·)$, $\chi_{175}(153,·)$, $\chi_{175}(27,·)$, $\chi_{175}(29,·)$, $\chi_{175}(34,·)$, $\chi_{175}(36,·)$, $\chi_{175}(6,·)$, $\chi_{175}(167,·)$, $\chi_{175}(41,·)$, $\chi_{175}(43,·)$, $\chi_{175}(174,·)$, $\chi_{175}(48,·)$, $\chi_{175}(57,·)$, $\chi_{175}(62,·)$, $\chi_{175}(64,·)$, $\chi_{175}(69,·)$, $\chi_{175}(71,·)$, $\chi_{175}(76,·)$, $\chi_{175}(162,·)$, $\chi_{175}(78,·)$, $\chi_{175}(141,·)$, $\chi_{175}(83,·)$, $\chi_{175}(92,·)$, $\chi_{175}(97,·)$, $\chi_{175}(99,·)$, $\chi_{175}(104,·)$, $\chi_{175}(106,·)$, $\chi_{175}(111,·)$, $\chi_{175}(113,·)$, $\chi_{175}(118,·)$, $\chi_{175}(169,·)$, $\chi_{175}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{4}a^{22}+\frac{1}{4}a^{17}+\frac{1}{4}a^{12}+\frac{1}{4}a^{7}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{23}-\frac{3}{8}a^{18}+\frac{1}{8}a^{13}-\frac{3}{8}a^{8}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{24}+\frac{5}{16}a^{19}-\frac{7}{16}a^{14}-\frac{3}{16}a^{9}+\frac{1}{16}a^{4}$, $\frac{1}{129568}a^{25}+\frac{5}{32}a^{20}+\frac{9}{32}a^{15}-\frac{3}{32}a^{10}+\frac{1}{32}a^{5}-\frac{627}{4049}$, $\frac{1}{259136}a^{26}+\frac{5}{64}a^{21}-\frac{23}{64}a^{16}+\frac{29}{64}a^{11}-\frac{31}{64}a^{6}+\frac{1711}{4049}a$, $\frac{1}{518272}a^{27}+\frac{5}{128}a^{22}+\frac{41}{128}a^{17}+\frac{29}{128}a^{12}+\frac{33}{128}a^{7}+\frac{1711}{8098}a^{2}$, $\frac{1}{1036544}a^{28}+\frac{5}{256}a^{23}-\frac{87}{256}a^{18}+\frac{29}{256}a^{13}-\frac{95}{256}a^{8}+\frac{1711}{16196}a^{3}$, $\frac{1}{2073088}a^{29}+\frac{5}{512}a^{24}-\frac{87}{512}a^{19}-\frac{227}{512}a^{14}+\frac{161}{512}a^{9}-\frac{14485}{32392}a^{4}$, $\frac{1}{4146176}a^{30}-\frac{11}{4146176}a^{25}-\frac{503}{1024}a^{20}+\frac{253}{1024}a^{15}+\frac{1}{1024}a^{10}-\frac{627}{129568}a^{5}+\frac{89}{4049}$, $\frac{1}{8292352}a^{31}-\frac{11}{8292352}a^{26}-\frac{503}{2048}a^{21}+\frac{253}{2048}a^{16}-\frac{1023}{2048}a^{11}+\frac{128941}{259136}a^{6}-\frac{1980}{4049}a$, $\frac{1}{16584704}a^{32}-\frac{11}{16584704}a^{27}-\frac{503}{4096}a^{22}-\frac{1795}{4096}a^{17}-\frac{1023}{4096}a^{12}-\frac{130195}{518272}a^{7}-\frac{990}{4049}a^{2}$, $\frac{1}{33169408}a^{33}-\frac{11}{33169408}a^{28}-\frac{503}{8192}a^{23}-\frac{1795}{8192}a^{18}+\frac{3073}{8192}a^{13}-\frac{130195}{1036544}a^{8}+\frac{3059}{8098}a^{3}$, $\frac{1}{66338816}a^{34}-\frac{11}{66338816}a^{29}-\frac{503}{16384}a^{24}-\frac{1795}{16384}a^{19}+\frac{3073}{16384}a^{14}-\frac{130195}{2073088}a^{9}-\frac{5039}{16196}a^{4}$, $\frac{1}{132677632}a^{35}-\frac{11}{132677632}a^{30}+\frac{89}{132677632}a^{25}+\frac{7421}{32768}a^{20}+\frac{1}{32768}a^{15}-\frac{627}{4146176}a^{10}+\frac{89}{129568}a^{5}-\frac{11}{4049}$, $\frac{1}{265355264}a^{36}-\frac{11}{265355264}a^{31}+\frac{89}{265355264}a^{26}+\frac{7421}{65536}a^{21}-\frac{32767}{65536}a^{16}+\frac{4145549}{8292352}a^{11}-\frac{129479}{259136}a^{6}+\frac{2019}{4049}a$, $\frac{1}{530710528}a^{37}-\frac{11}{530710528}a^{32}+\frac{89}{530710528}a^{27}+\frac{7421}{131072}a^{22}+\frac{32769}{131072}a^{17}+\frac{4145549}{16584704}a^{12}+\frac{129657}{518272}a^{7}+\frac{2019}{8098}a^{2}$, $\frac{1}{1061421056}a^{38}-\frac{11}{1061421056}a^{33}+\frac{89}{1061421056}a^{28}+\frac{7421}{262144}a^{23}+\frac{32769}{262144}a^{18}-\frac{12439155}{33169408}a^{13}+\frac{129657}{1036544}a^{8}-\frac{6079}{16196}a^{3}$, $\frac{1}{2122842112}a^{39}-\frac{11}{2122842112}a^{34}+\frac{89}{2122842112}a^{29}+\frac{7421}{524288}a^{24}-\frac{229375}{524288}a^{19}-\frac{12439155}{66338816}a^{14}+\frac{129657}{2073088}a^{9}+\frac{10117}{32392}a^{4}$ 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:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1}{33169408} a^{38} - \frac{364229}{33169408} a^{13} \)  (order $50$) 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^40 - 11*x^35 + 89*x^30 - 627*x^25 + 4049*x^20 - 20064*x^15 + 91136*x^10 - 360448*x^5 + 1048576)
 
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^40 - 11*x^35 + 89*x^30 - 627*x^25 + 4049*x^20 - 20064*x^15 + 91136*x^10 - 360448*x^5 + 1048576, 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^40 - 11*x^35 + 89*x^30 - 627*x^25 + 4049*x^20 - 20064*x^15 + 91136*x^10 - 360448*x^5 + 1048576);
 
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^40 - 11*x^35 + 89*x^30 - 627*x^25 + 4049*x^20 - 20064*x^15 + 91136*x^10 - 360448*x^5 + 1048576);
 
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

$C_2\times C_{20}$ (as 40T2):

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 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\zeta_{5})\), 4.4.6125.1, 5.5.390625.1, 8.0.37515625.1, 10.0.12822723388671875.1, \(\Q(\zeta_{25})^+\), 10.0.2564544677734375.1, 20.0.164422235102392733097076416015625.1, \(\Q(\zeta_{25})\), 20.20.822111175511963665485382080078125.1

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 $20^{2}$ $20^{2}$ R R ${\href{/padicField/11.5.0.1}{5} }^{8}$ $20^{2}$ $20^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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 Deg $40$$20$$2$$70$
\(7\) Copy content Toggle raw display 7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$