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Properties

Label 40.0.21620507387...0000.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 5^{30}\cdot 11^{32}$
Root discriminant $45.54$
Ramified primes $2, 5, 11$
Class number $155$ (GRH)
Class group $[155]$ (GRH)
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -15, 0, 190, 0, -2353, 0, 29056, 0, -72116, 0, 128969, 0, -203056, 0, 292022, 0, -264514, 0, 193618, 0, -124361, 0, 70035, 0, -30590, 0, 11685, 0, -3971, 0, 1156, 0, -260, 0, 53, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 9*x^38 + 53*x^36 - 260*x^34 + 1156*x^32 - 3971*x^30 + 11685*x^28 - 30590*x^26 + 70035*x^24 - 124361*x^22 + 193618*x^20 - 264514*x^18 + 292022*x^16 - 203056*x^14 + 128969*x^12 - 72116*x^10 + 29056*x^8 - 2353*x^6 + 190*x^4 - 15*x^2 + 1)
 
gp: K = bnfinit(x^40 - 9*x^38 + 53*x^36 - 260*x^34 + 1156*x^32 - 3971*x^30 + 11685*x^28 - 30590*x^26 + 70035*x^24 - 124361*x^22 + 193618*x^20 - 264514*x^18 + 292022*x^16 - 203056*x^14 + 128969*x^12 - 72116*x^10 + 29056*x^8 - 2353*x^6 + 190*x^4 - 15*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{40} - 9 x^{38} + 53 x^{36} - 260 x^{34} + 1156 x^{32} - 3971 x^{30} + 11685 x^{28} - 30590 x^{26} + 70035 x^{24} - 124361 x^{22} + 193618 x^{20} - 264514 x^{18} + 292022 x^{16} - 203056 x^{14} + 128969 x^{12} - 72116 x^{10} + 29056 x^{8} - 2353 x^{6} + 190 x^{4} - 15 x^{2} + 1 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2162050738724101412398710020310959104000000000000000000000000000000=2^{40}\cdot 5^{30}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.54$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(220=2^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(3,·)$, $\chi_{220}(133,·)$, $\chi_{220}(9,·)$, $\chi_{220}(141,·)$, $\chi_{220}(147,·)$, $\chi_{220}(23,·)$, $\chi_{220}(27,·)$, $\chi_{220}(157,·)$, $\chi_{220}(159,·)$, $\chi_{220}(163,·)$, $\chi_{220}(37,·)$, $\chi_{220}(177,·)$, $\chi_{220}(169,·)$, $\chi_{220}(71,·)$, $\chi_{220}(47,·)$, $\chi_{220}(49,·)$, $\chi_{220}(179,·)$, $\chi_{220}(181,·)$, $\chi_{220}(137,·)$, $\chi_{220}(31,·)$, $\chi_{220}(53,·)$, $\chi_{220}(67,·)$, $\chi_{220}(69,·)$, $\chi_{220}(199,·)$, $\chi_{220}(201,·)$, $\chi_{220}(203,·)$, $\chi_{220}(207,·)$, $\chi_{220}(81,·)$, $\chi_{220}(213,·)$, $\chi_{220}(89,·)$, $\chi_{220}(91,·)$, $\chi_{220}(93,·)$, $\chi_{220}(97,·)$, $\chi_{220}(59,·)$, $\chi_{220}(103,·)$, $\chi_{220}(111,·)$, $\chi_{220}(113,·)$, $\chi_{220}(119,·)$, $\chi_{220}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{234425411086737421} a^{34} + \frac{13518200052769406}{234425411086737421} a^{32} - \frac{65833524192462789}{234425411086737421} a^{30} - \frac{20433294832115723}{234425411086737421} a^{28} - \frac{86876548576091573}{234425411086737421} a^{26} + \frac{101344618896966894}{234425411086737421} a^{24} - \frac{87435839101689013}{234425411086737421} a^{22} - \frac{4187296534272345}{234425411086737421} a^{20} + \frac{27983614648654076}{234425411086737421} a^{18} + \frac{85800938282018068}{234425411086737421} a^{16} + \frac{42104619363901185}{234425411086737421} a^{14} - \frac{89443189315860660}{234425411086737421} a^{12} + \frac{114511635870379700}{234425411086737421} a^{10} + \frac{91697677915639135}{234425411086737421} a^{8} + \frac{22514369310365389}{234425411086737421} a^{6} + \frac{94466978223430401}{234425411086737421} a^{4} - \frac{39955723921154417}{234425411086737421} a^{2} + \frac{73884285512829285}{234425411086737421}$, $\frac{1}{234425411086737421} a^{35} + \frac{13518200052769406}{234425411086737421} a^{33} - \frac{65833524192462789}{234425411086737421} a^{31} - \frac{20433294832115723}{234425411086737421} a^{29} - \frac{86876548576091573}{234425411086737421} a^{27} + \frac{101344618896966894}{234425411086737421} a^{25} - \frac{87435839101689013}{234425411086737421} a^{23} - \frac{4187296534272345}{234425411086737421} a^{21} + \frac{27983614648654076}{234425411086737421} a^{19} + \frac{85800938282018068}{234425411086737421} a^{17} + \frac{42104619363901185}{234425411086737421} a^{15} - \frac{89443189315860660}{234425411086737421} a^{13} + \frac{114511635870379700}{234425411086737421} a^{11} + \frac{91697677915639135}{234425411086737421} a^{9} + \frac{22514369310365389}{234425411086737421} a^{7} + \frac{94466978223430401}{234425411086737421} a^{5} - \frac{39955723921154417}{234425411086737421} a^{3} + \frac{73884285512829285}{234425411086737421} a$, $\frac{1}{234425411086737421} a^{36} - \frac{68754752360449093}{234425411086737421} a^{32} + \frac{77128489216892857}{234425411086737421} a^{30} + \frac{58849607706851771}{234425411086737421} a^{28} + \frac{113901823037097024}{234425411086737421} a^{26} - \frac{67238106345621796}{234425411086737421} a^{24} + \frac{47267567493151067}{234425411086737421} a^{22} + \frac{51231680090722026}{234425411086737421} a^{20} - \frac{26493186357585276}{234425411086737421} a^{18} - \frac{73659650091367177}{234425411086737421} a^{16} - \frac{32311846197611962}{234425411086737421} a^{14} - \frac{53504726083231895}{234425411086737421} a^{12} + \frac{38860356124084795}{234425411086737421} a^{10} - \frac{105689486442819809}{234425411086737421} a^{8} - \frac{103350495725509894}{234425411086737421} a^{6} + \frac{72194495950593475}{234425411086737421} a^{4} - \frac{48637259237983121}{234425411086737421} a^{2} - \frac{75682692523940347}{234425411086737421}$, $\frac{1}{234425411086737421} a^{37} - \frac{68754752360449093}{234425411086737421} a^{33} + \frac{77128489216892857}{234425411086737421} a^{31} + \frac{58849607706851771}{234425411086737421} a^{29} + \frac{113901823037097024}{234425411086737421} a^{27} - \frac{67238106345621796}{234425411086737421} a^{25} + \frac{47267567493151067}{234425411086737421} a^{23} + \frac{51231680090722026}{234425411086737421} a^{21} - \frac{26493186357585276}{234425411086737421} a^{19} - \frac{73659650091367177}{234425411086737421} a^{17} - \frac{32311846197611962}{234425411086737421} a^{15} - \frac{53504726083231895}{234425411086737421} a^{13} + \frac{38860356124084795}{234425411086737421} a^{11} - \frac{105689486442819809}{234425411086737421} a^{9} - \frac{103350495725509894}{234425411086737421} a^{7} + \frac{72194495950593475}{234425411086737421} a^{5} - \frac{48637259237983121}{234425411086737421} a^{3} - \frac{75682692523940347}{234425411086737421} a$, $\frac{1}{234425411086737421} a^{38} + \frac{90192232013913741}{234425411086737421} a^{32} + \frac{107979966750978021}{234425411086737421} a^{30} - \frac{52942195164755760}{234425411086737421} a^{28} - \frac{23291812546524549}{234425411086737421} a^{26} - \frac{67782684854137782}{234425411086737421} a^{24} + \frac{95361832283953612}{234425411086737421} a^{22} - \frac{45444526576961110}{234425411086737421} a^{20} + \frac{51412482975660505}{234425411086737421} a^{18} - \frac{21220530837732928}{234425411086737421} a^{16} + \frac{104104995549247932}{234425411086737421} a^{14} - \frac{75423139788223966}{234425411086737421} a^{12} - \frac{11604721202771412}{234425411086737421} a^{10} - \frac{41576451642245981}{234425411086737421} a^{8} - \frac{86907643915982136}{234425411086737421} a^{6} + \frac{14306170445464466}{234425411086737421} a^{4} - \frac{69844591026167126}{234425411086737421} a^{2} + \frac{73505267368585198}{234425411086737421}$, $\frac{1}{234425411086737421} a^{39} + \frac{90192232013913741}{234425411086737421} a^{33} + \frac{107979966750978021}{234425411086737421} a^{31} - \frac{52942195164755760}{234425411086737421} a^{29} - \frac{23291812546524549}{234425411086737421} a^{27} - \frac{67782684854137782}{234425411086737421} a^{25} + \frac{95361832283953612}{234425411086737421} a^{23} - \frac{45444526576961110}{234425411086737421} a^{21} + \frac{51412482975660505}{234425411086737421} a^{19} - \frac{21220530837732928}{234425411086737421} a^{17} + \frac{104104995549247932}{234425411086737421} a^{15} - \frac{75423139788223966}{234425411086737421} a^{13} - \frac{11604721202771412}{234425411086737421} a^{11} - \frac{41576451642245981}{234425411086737421} a^{9} - \frac{86907643915982136}{234425411086737421} a^{7} + \frac{14306170445464466}{234425411086737421} a^{5} - \frac{69844591026167126}{234425411086737421} a^{3} + \frac{73505267368585198}{234425411086737421} a$

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

Class group and class number

$C_{155}$, which has order $155$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1033956405789076}{234425411086737421} a^{39} - \frac{8013162144865339}{234425411086737421} a^{37} + \frac{43721585159080928}{234425411086737421} a^{35} - \frac{205277567524804009}{234425411086737421} a^{33} + \frac{888242406601801218}{234425411086737421} a^{31} - \frac{2753832105775726525}{234425411086737421} a^{29} + \frac{7580192899941163425}{234425411086737421} a^{27} - \frac{18683370690521648518}{234425411086737421} a^{25} + \frac{39202718867588571471}{234425411086737421} a^{23} - \frac{54579899916561864416}{234425411086737421} a^{21} + \frac{77125910177024546068}{234425411086737421} a^{19} - \frac{89553253780290405314}{234425411086737421} a^{17} + \frac{62721826560962435845}{234425411086737421} a^{15} + \frac{28066822999132149804}{234425411086737421} a^{13} + \frac{22894305861155630396}{234425411086737421} a^{11} - \frac{9569672732737315517}{234425411086737421} a^{9} + \frac{774987253153404929}{234425411086737421} a^{7} - \frac{62591289564731565}{234425411086737421} a^{5} + \frac{10171515181693540290}{234425411086737421} a^{3} - \frac{332343130432203}{234425411086737421} a \) (order $20$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2347083738648341.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{20})\), 10.0.219503494144.1, 10.10.669871503125.1, 10.0.685948419200000.1, 20.0.470525233802978928640000000000.1, 20.0.1402274470934209014892578125.1, 20.20.1470391355634309152000000000000000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20^{2}$ R $20^{2}$ R $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$