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Properties

Label 40.0.130...936.7
Degree $40$
Signature $[0, 20]$
Discriminant $1.303\times 10^{71}$
Root discriminant \(59.96\)
Ramified primes $2,3,11$
Class number $341000$ (GRH)
Class group [5, 10, 6820] (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 40*x^38 + 741*x^36 + 8436*x^34 + 66044*x^32 + 376960*x^30 + 1622694*x^28 + 5375528*x^26 + 13860054*x^24 + 27947920*x^22 + 44043506*x^20 + 53927016*x^18 + 50713585*x^16 + 35964944*x^14 + 18713229*x^12 + 6857500*x^10 + 1662386*x^8 + 240976*x^6 + 17560*x^4 + 480*x^2 + 1)
 
gp: K = bnfinit(y^40 + 40*y^38 + 741*y^36 + 8436*y^34 + 66044*y^32 + 376960*y^30 + 1622694*y^28 + 5375528*y^26 + 13860054*y^24 + 27947920*y^22 + 44043506*y^20 + 53927016*y^18 + 50713585*y^16 + 35964944*y^14 + 18713229*y^12 + 6857500*y^10 + 1662386*y^8 + 240976*y^6 + 17560*y^4 + 480*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 40*x^38 + 741*x^36 + 8436*x^34 + 66044*x^32 + 376960*x^30 + 1622694*x^28 + 5375528*x^26 + 13860054*x^24 + 27947920*x^22 + 44043506*x^20 + 53927016*x^18 + 50713585*x^16 + 35964944*x^14 + 18713229*x^12 + 6857500*x^10 + 1662386*x^8 + 240976*x^6 + 17560*x^4 + 480*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 40*x^38 + 741*x^36 + 8436*x^34 + 66044*x^32 + 376960*x^30 + 1622694*x^28 + 5375528*x^26 + 13860054*x^24 + 27947920*x^22 + 44043506*x^20 + 53927016*x^18 + 50713585*x^16 + 35964944*x^14 + 18713229*x^12 + 6857500*x^10 + 1662386*x^8 + 240976*x^6 + 17560*x^4 + 480*x^2 + 1)
 

\( x^{40} + 40 x^{38} + 741 x^{36} + 8436 x^{34} + 66044 x^{32} + 376960 x^{30} + 1622694 x^{28} + 5375528 x^{26} + \cdots + 1 \) 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:   \(130305099804548492884220428175380349368393046678311823693003457545895936\) \(\medspace = 2^{80}\cdot 3^{20}\cdot 11^{36}\) 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:  \(59.96\)
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:  $2^{2}3^{1/2}11^{9/10}\approx 59.96171354565991$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(5,·)$, $\chi_{264}(7,·)$, $\chi_{264}(151,·)$, $\chi_{264}(13,·)$, $\chi_{264}(17,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(47,·)$, $\chi_{264}(161,·)$, $\chi_{264}(35,·)$, $\chi_{264}(163,·)$, $\chi_{264}(41,·)$, $\chi_{264}(175,·)$, $\chi_{264}(49,·)$, $\chi_{264}(53,·)$, $\chi_{264}(61,·)$, $\chi_{264}(191,·)$, $\chi_{264}(65,·)$, $\chi_{264}(67,·)$, $\chi_{264}(71,·)$, $\chi_{264}(119,·)$, $\chi_{264}(205,·)$, $\chi_{264}(79,·)$, $\chi_{264}(83,·)$, $\chi_{264}(85,·)$, $\chi_{264}(91,·)$, $\chi_{264}(221,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(233,·)$, $\chi_{264}(107,·)$, $\chi_{264}(109,·)$, $\chi_{264}(115,·)$, $\chi_{264}(235,·)$, $\chi_{264}(245,·)$, $\chi_{264}(169,·)$, $\chi_{264}(125,·)$, $\chi_{264}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

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

$C_{5}\times C_{10}\times C_{6820}$, which has order $341000$ (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:  $19$
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:   $a^{2}+2$, $2a^{38}+76a^{36}+1331a^{34}+14246a^{32}+104190a^{30}+551492a^{28}+2182858a^{26}+6582628a^{24}+15267031a^{22}+27296698a^{20}+37477102a^{18}+39106444a^{16}+30468217a^{14}+17242398a^{12}+6796891a^{10}+1748702a^{8}+263969a^{6}+19430a^{4}+521a^{2}+2$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24751a^{12}+27444a^{10}+19251a^{8}+7896a^{6}+1611a^{4}+108a^{2}+1$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24752a^{12}+27456a^{10}+19305a^{8}+8008a^{6}+1716a^{4}+144a^{2}+2$, $a^{12}+12a^{10}+54a^{8}+112a^{6}+105a^{4}+36a^{2}+2$, $a^{30}+30a^{28}+405a^{26}+3250a^{24}+17250a^{22}+63756a^{20}+168245a^{18}+319770a^{16}+436049a^{14}+419886a^{12}+277057a^{10}+119130a^{8}+30646a^{6}+4004a^{4}+176a^{2}$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273296a^{28}+1076103a^{26}+3223350a^{24}+7413705a^{22}+13123110a^{20}+17809935a^{18}+18349630a^{16}+14115100a^{14}+7904456a^{12}+3105322a^{10}+810084a^{8}+128877a^{6}+10830a^{4}+361a^{2}+2$, $a^{10}+10a^{8}+35a^{6}+50a^{4}+25a^{2}+2$, $a^{26}+26a^{24}+299a^{22}+2002a^{20}+8645a^{18}+25194a^{16}+50388a^{14}+68952a^{12}+63206a^{10}+37180a^{8}+13013a^{6}+2366a^{4}+169a^{2}+2$, $a^{39}+38a^{37}+665a^{35}+7105a^{33}+51798a^{31}+272770a^{29}+1071202a^{27}+3192670a^{25}+7277426a^{23}+12680889a^{21}+16746808a^{19}+16449914a^{17}+11607141a^{15}+5496727a^{13}+1470831a^{11}+60609a^{9}-86316a^{7}-22993a^{5}-1870a^{3}-41a$, $a$, $a^{39}+38a^{37}+665a^{35}+7105a^{33}+51798a^{31}+272770a^{29}+1071202a^{27}+3192670a^{25}+7277426a^{23}+12680889a^{21}+16746808a^{19}+16449914a^{17}+11607141a^{15}+5496727a^{13}+1470831a^{11}+60609a^{9}-86316a^{7}-22993a^{5}-1870a^{3}-40a$, $a^{37}+37a^{35}+629a^{33}+6512a^{31}+45880a^{29}+232841a^{27}+878787a^{25}+2510820a^{23}+5476185a^{21}+9126975a^{19}+11560835a^{17}+10994920a^{15}+7696444a^{13}+3848222a^{11}+1314610a^{9}+286824a^{7}+35853a^{5}+2109a^{3}+37a$, $a^{7}+7a^{5}+14a^{3}+7a$, $a^{27}+27a^{25}+324a^{23}+2277a^{21}+10395a^{19}+32320a^{17}+69785a^{15}+104771a^{13}+107848a^{11}+73865a^{9}+32010a^{7}+8085a^{5}+1023a^{3}+44a$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273296a^{28}+1076102a^{26}+3223324a^{24}+7413405a^{22}+13121086a^{20}+17801081a^{18}+18323314a^{16}+14060973a^{14}+7827510a^{12}+3031183a^{10}+763687a^{8}+111481a^{6}+7514a^{4}+151a^{2}$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273296a^{28}+1076103a^{26}+3223350a^{24}+7413705a^{22}+13123110a^{20}+17809935a^{18}+18349630a^{16}+14115100a^{14}+7904456a^{12}+3105322a^{10}+810084a^{8}+128877a^{6}+10830a^{4}+361a^{2}+1$, $a^{19}+19a^{17}+152a^{15}+665a^{13}+1729a^{11}+2717a^{9}+2508a^{7}+1254a^{5}+285a^{3}+19a$, $a^{11}+11a^{9}+44a^{7}+77a^{5}+55a^{3}+11a$ 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:  \( 45016485591237.79 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 45016485591237.79 \cdot 341000}{2\cdot\sqrt{130305099804548492884220428175380349368393046678311823693003457545895936}}\cr\approx \mathstrut & 0.195529653554255 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^40 + 40*x^38 + 741*x^36 + 8436*x^34 + 66044*x^32 + 376960*x^30 + 1622694*x^28 + 5375528*x^26 + 13860054*x^24 + 27947920*x^22 + 44043506*x^20 + 53927016*x^18 + 50713585*x^16 + 35964944*x^14 + 18713229*x^12 + 6857500*x^10 + 1662386*x^8 + 240976*x^6 + 17560*x^4 + 480*x^2 + 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^40 + 40*x^38 + 741*x^36 + 8436*x^34 + 66044*x^32 + 376960*x^30 + 1622694*x^28 + 5375528*x^26 + 13860054*x^24 + 27947920*x^22 + 44043506*x^20 + 53927016*x^18 + 50713585*x^16 + 35964944*x^14 + 18713229*x^12 + 6857500*x^10 + 1662386*x^8 + 240976*x^6 + 17560*x^4 + 480*x^2 + 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^40 + 40*x^38 + 741*x^36 + 8436*x^34 + 66044*x^32 + 376960*x^30 + 1622694*x^28 + 5375528*x^26 + 13860054*x^24 + 27947920*x^22 + 44043506*x^20 + 53927016*x^18 + 50713585*x^16 + 35964944*x^14 + 18713229*x^12 + 6857500*x^10 + 1662386*x^8 + 240976*x^6 + 17560*x^4 + 480*x^2 + 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^40 + 40*x^38 + 741*x^36 + 8436*x^34 + 66044*x^32 + 376960*x^30 + 1622694*x^28 + 5375528*x^26 + 13860054*x^24 + 27947920*x^22 + 44043506*x^20 + 53927016*x^18 + 50713585*x^16 + 35964944*x^14 + 18713229*x^12 + 6857500*x^10 + 1662386*x^8 + 240976*x^6 + 17560*x^4 + 480*x^2 + 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

$C_2^2\times C_{10}$ (as 40T7):

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 $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-22}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6}, \sqrt{-22})\), \(\Q(\sqrt{-2}, \sqrt{11})\), \(\Q(\sqrt{3}, \sqrt{-22})\), \(\Q(\sqrt{-2}, \sqrt{33})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.0.77720518656.4, 10.0.77265229938688.1, \(\Q(\zeta_{33})^+\), 10.0.1706859170463744.1, 10.0.7024111812608.1, \(\Q(\zeta_{44})^+\), 10.0.18775450875101184.1, 10.10.53339349076992.1, 20.0.352517555563337816067682238201856.2, 20.0.6113193735657808322804901216256.3, 20.0.360977976896857923653306611918700544.1, 20.0.352517555563337816067682238201856.7, \(\Q(\zeta_{132})^+\), 20.0.2983289065263288625233938941476864.1, 20.0.360977976896857923653306611918700544.3

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 R ${\href{/padicField/5.10.0.1}{10} }^{4}$ ${\href{/padicField/7.10.0.1}{10} }^{4}$ R ${\href{/padicField/13.10.0.1}{10} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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
\(2\) Copy content Toggle raw display Deg $20$$4$$5$$40$
Deg $20$$4$$5$$40$
\(3\) Copy content Toggle raw display 3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
\(11\) Copy content Toggle raw display 11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$