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Properties

Label 44.0.11724385642...0704.1
Degree $44$
Signature $[0, 22]$
Discriminant $2^{44}\cdot 89^{43}$
Root discriminant $160.74$
Ramified primes $2, 89$
Class number Not computed
Class group Not computed
Galois group $C_{44}$ (as 44T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, 0, 13617, 0, 791922, 0, 22464757, 0, 329351798, 0, 2520532510, 0, 11291273538, 0, 32223471809, 0, 61906589562, 0, 83072278176, 0, 79915633830, 0, 56180587217, 0, 29270077030, 0, 11410780513, 0, 3346368138, 0, 738903721, 0, 122342159, 0, 15033079, 0, 1345057, 0, 84817, 0, 3560, 0, 89, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 89*x^42 + 3560*x^40 + 84817*x^38 + 1345057*x^36 + 15033079*x^34 + 122342159*x^32 + 738903721*x^30 + 3346368138*x^28 + 11410780513*x^26 + 29270077030*x^24 + 56180587217*x^22 + 79915633830*x^20 + 83072278176*x^18 + 61906589562*x^16 + 32223471809*x^14 + 11291273538*x^12 + 2520532510*x^10 + 329351798*x^8 + 22464757*x^6 + 791922*x^4 + 13617*x^2 + 89)
 
gp: K = bnfinit(x^44 + 89*x^42 + 3560*x^40 + 84817*x^38 + 1345057*x^36 + 15033079*x^34 + 122342159*x^32 + 738903721*x^30 + 3346368138*x^28 + 11410780513*x^26 + 29270077030*x^24 + 56180587217*x^22 + 79915633830*x^20 + 83072278176*x^18 + 61906589562*x^16 + 32223471809*x^14 + 11291273538*x^12 + 2520532510*x^10 + 329351798*x^8 + 22464757*x^6 + 791922*x^4 + 13617*x^2 + 89, 1)
 

Normalized defining polynomial

\( x^{44} + 89 x^{42} + 3560 x^{40} + 84817 x^{38} + 1345057 x^{36} + 15033079 x^{34} + 122342159 x^{32} + 738903721 x^{30} + 3346368138 x^{28} + 11410780513 x^{26} + 29270077030 x^{24} + 56180587217 x^{22} + 79915633830 x^{20} + 83072278176 x^{18} + 61906589562 x^{16} + 32223471809 x^{14} + 11291273538 x^{12} + 2520532510 x^{10} + 329351798 x^{8} + 22464757 x^{6} + 791922 x^{4} + 13617 x^{2} + 89 \)

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

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 22]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11724385642028745774656376755923007752612263949364698259094951647151017819768316744951538808520704=2^{44}\cdot 89^{43}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $160.74$
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:  $2, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(356=2^{2}\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{356}(1,·)$, $\chi_{356}(131,·)$, $\chi_{356}(133,·)$, $\chi_{356}(265,·)$, $\chi_{356}(269,·)$, $\chi_{356}(153,·)$, $\chi_{356}(25,·)$, $\chi_{356}(47,·)$, $\chi_{356}(287,·)$, $\chi_{356}(289,·)$, $\chi_{356}(71,·)$, $\chi_{356}(45,·)$, $\chi_{356}(303,·)$, $\chi_{356}(177,·)$, $\chi_{356}(307,·)$, $\chi_{356}(55,·)$, $\chi_{356}(57,·)$, $\chi_{356}(187,·)$, $\chi_{356}(189,·)$, $\chi_{356}(195,·)$, $\chi_{356}(199,·)$, $\chi_{356}(73,·)$, $\chi_{356}(183,·)$, $\chi_{356}(79,·)$, $\chi_{356}(81,·)$, $\chi_{356}(335,·)$, $\chi_{356}(99,·)$, $\chi_{356}(85,·)$, $\chi_{356}(217,·)$, $\chi_{356}(347,·)$, $\chi_{356}(93,·)$, $\chi_{356}(351,·)$, $\chi_{356}(97,·)$, $\chi_{356}(227,·)$, $\chi_{356}(231,·)$, $\chi_{356}(105,·)$, $\chi_{356}(107,·)$, $\chi_{356}(317,·)$, $\chi_{356}(339,·)$, $\chi_{356}(245,·)$, $\chi_{356}(247,·)$, $\chi_{356}(121,·)$, $\chi_{356}(123,·)$, $\chi_{356}(345,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{179} a^{36} + \frac{25}{179} a^{34} - \frac{29}{179} a^{32} + \frac{33}{179} a^{30} - \frac{20}{179} a^{28} - \frac{1}{179} a^{26} + \frac{4}{179} a^{24} + \frac{58}{179} a^{22} + \frac{62}{179} a^{20} - \frac{51}{179} a^{18} - \frac{3}{179} a^{16} + \frac{42}{179} a^{14} - \frac{2}{179} a^{12} - \frac{11}{179} a^{10} - \frac{12}{179} a^{8} - \frac{24}{179} a^{6} + \frac{18}{179} a^{4} + \frac{77}{179} a^{2} + \frac{85}{179}$, $\frac{1}{179} a^{37} + \frac{25}{179} a^{35} - \frac{29}{179} a^{33} + \frac{33}{179} a^{31} - \frac{20}{179} a^{29} - \frac{1}{179} a^{27} + \frac{4}{179} a^{25} + \frac{58}{179} a^{23} + \frac{62}{179} a^{21} - \frac{51}{179} a^{19} - \frac{3}{179} a^{17} + \frac{42}{179} a^{15} - \frac{2}{179} a^{13} - \frac{11}{179} a^{11} - \frac{12}{179} a^{9} - \frac{24}{179} a^{7} + \frac{18}{179} a^{5} + \frac{77}{179} a^{3} + \frac{85}{179} a$, $\frac{1}{179} a^{38} + \frac{62}{179} a^{34} + \frac{42}{179} a^{32} + \frac{50}{179} a^{30} - \frac{38}{179} a^{28} + \frac{29}{179} a^{26} - \frac{42}{179} a^{24} + \frac{44}{179} a^{22} + \frac{10}{179} a^{20} + \frac{19}{179} a^{18} - \frac{62}{179} a^{16} + \frac{22}{179} a^{14} + \frac{39}{179} a^{12} + \frac{84}{179} a^{10} - \frac{82}{179} a^{8} + \frac{81}{179} a^{6} - \frac{15}{179} a^{4} - \frac{50}{179} a^{2} + \frac{23}{179}$, $\frac{1}{179} a^{39} + \frac{62}{179} a^{35} + \frac{42}{179} a^{33} + \frac{50}{179} a^{31} - \frac{38}{179} a^{29} + \frac{29}{179} a^{27} - \frac{42}{179} a^{25} + \frac{44}{179} a^{23} + \frac{10}{179} a^{21} + \frac{19}{179} a^{19} - \frac{62}{179} a^{17} + \frac{22}{179} a^{15} + \frac{39}{179} a^{13} + \frac{84}{179} a^{11} - \frac{82}{179} a^{9} + \frac{81}{179} a^{7} - \frac{15}{179} a^{5} - \frac{50}{179} a^{3} + \frac{23}{179} a$, $\frac{1}{85741} a^{40} - \frac{50}{85741} a^{38} + \frac{233}{85741} a^{36} - \frac{22411}{85741} a^{34} + \frac{21273}{85741} a^{32} + \frac{23332}{85741} a^{30} + \frac{29834}{85741} a^{28} - \frac{20458}{85741} a^{26} - \frac{16504}{85741} a^{24} - \frac{8740}{85741} a^{22} + \frac{15849}{85741} a^{20} - \frac{14387}{85741} a^{18} + \frac{21941}{85741} a^{16} + \frac{4510}{85741} a^{14} - \frac{38366}{85741} a^{12} - \frac{41426}{85741} a^{10} + \frac{12153}{85741} a^{8} + \frac{21545}{85741} a^{6} - \frac{19671}{85741} a^{4} - \frac{34430}{85741} a^{2} - \frac{30649}{85741}$, $\frac{1}{85741} a^{41} - \frac{50}{85741} a^{39} + \frac{233}{85741} a^{37} - \frac{22411}{85741} a^{35} + \frac{21273}{85741} a^{33} + \frac{23332}{85741} a^{31} + \frac{29834}{85741} a^{29} - \frac{20458}{85741} a^{27} - \frac{16504}{85741} a^{25} - \frac{8740}{85741} a^{23} + \frac{15849}{85741} a^{21} - \frac{14387}{85741} a^{19} + \frac{21941}{85741} a^{17} + \frac{4510}{85741} a^{15} - \frac{38366}{85741} a^{13} - \frac{41426}{85741} a^{11} + \frac{12153}{85741} a^{9} + \frac{21545}{85741} a^{7} - \frac{19671}{85741} a^{5} - \frac{34430}{85741} a^{3} - \frac{30649}{85741} a$, $\frac{1}{1773725647879019262913712293886257344128658191130053474400713931727} a^{42} - \frac{7678808732568459060819074787799734437267543194865281488869174}{1773725647879019262913712293886257344128658191130053474400713931727} a^{40} + \frac{1117354887762076870748036239669850930527936157001004395745591052}{1773725647879019262913712293886257344128658191130053474400713931727} a^{38} + \frac{4229697727873115951116815038860055973567279322103878852573601583}{1773725647879019262913712293886257344128658191130053474400713931727} a^{36} - \frac{292769922870197489005933941697121321084828900948687283839139675112}{1773725647879019262913712293886257344128658191130053474400713931727} a^{34} + \frac{8462902383583051320998665931845013268935327054054477488452516041}{1773725647879019262913712293886257344128658191130053474400713931727} a^{32} - \frac{719886799520751357934814465413917347013901807261399462761095201762}{1773725647879019262913712293886257344128658191130053474400713931727} a^{30} + \frac{616104327738972892838947796918866779810869496173014469384234217719}{1773725647879019262913712293886257344128658191130053474400713931727} a^{28} + \frac{116469693112723786488205258454063985926403538530184229460253161146}{1773725647879019262913712293886257344128658191130053474400713931727} a^{26} - \frac{587253741372928811585254085028572481093406182449636910125090514596}{1773725647879019262913712293886257344128658191130053474400713931727} a^{24} - \frac{843203378443540942019081531869901027396191085525746520280752380314}{1773725647879019262913712293886257344128658191130053474400713931727} a^{22} - \frac{470074263924110059085555896345284207138557596355776591023917136580}{1773725647879019262913712293886257344128658191130053474400713931727} a^{20} - \frac{209451347052866603393544052906702313536013039859360207726472835991}{1773725647879019262913712293886257344128658191130053474400713931727} a^{18} - \frac{413880775950260063962665482814941136723804069446498975883733751758}{1773725647879019262913712293886257344128658191130053474400713931727} a^{16} - \frac{875241438996746698457995530676084913500465935931621083853833594497}{1773725647879019262913712293886257344128658191130053474400713931727} a^{14} + \frac{60466585409600690161884940554067928522232302136838937931661601715}{1773725647879019262913712293886257344128658191130053474400713931727} a^{12} - \frac{485726025494363276618532225177901328507276159276176678054676853964}{1773725647879019262913712293886257344128658191130053474400713931727} a^{10} - \frac{165959056844894622769943648680324149238471581618850397529733554640}{1773725647879019262913712293886257344128658191130053474400713931727} a^{8} + \frac{859689877861723093023853377449202712707409616825890819515600280056}{1773725647879019262913712293886257344128658191130053474400713931727} a^{6} + \frac{396192289210241788563947865445458855817839968817628718819661083791}{1773725647879019262913712293886257344128658191130053474400713931727} a^{4} + \frac{752861951381189501111726070124087350492556649186302928822363945813}{1773725647879019262913712293886257344128658191130053474400713931727} a^{2} - \frac{82225258859283191951777519920807440004636171881109873738133652257}{1773725647879019262913712293886257344128658191130053474400713931727}$, $\frac{1}{1773725647879019262913712293886257344128658191130053474400713931727} a^{43} - \frac{7678808732568459060819074787799734437267543194865281488869174}{1773725647879019262913712293886257344128658191130053474400713931727} a^{41} + \frac{1117354887762076870748036239669850930527936157001004395745591052}{1773725647879019262913712293886257344128658191130053474400713931727} a^{39} + \frac{4229697727873115951116815038860055973567279322103878852573601583}{1773725647879019262913712293886257344128658191130053474400713931727} a^{37} - \frac{292769922870197489005933941697121321084828900948687283839139675112}{1773725647879019262913712293886257344128658191130053474400713931727} a^{35} + \frac{8462902383583051320998665931845013268935327054054477488452516041}{1773725647879019262913712293886257344128658191130053474400713931727} a^{33} - \frac{719886799520751357934814465413917347013901807261399462761095201762}{1773725647879019262913712293886257344128658191130053474400713931727} a^{31} + \frac{616104327738972892838947796918866779810869496173014469384234217719}{1773725647879019262913712293886257344128658191130053474400713931727} a^{29} + \frac{116469693112723786488205258454063985926403538530184229460253161146}{1773725647879019262913712293886257344128658191130053474400713931727} a^{27} - \frac{587253741372928811585254085028572481093406182449636910125090514596}{1773725647879019262913712293886257344128658191130053474400713931727} a^{25} - \frac{843203378443540942019081531869901027396191085525746520280752380314}{1773725647879019262913712293886257344128658191130053474400713931727} a^{23} - \frac{470074263924110059085555896345284207138557596355776591023917136580}{1773725647879019262913712293886257344128658191130053474400713931727} a^{21} - \frac{209451347052866603393544052906702313536013039859360207726472835991}{1773725647879019262913712293886257344128658191130053474400713931727} a^{19} - \frac{413880775950260063962665482814941136723804069446498975883733751758}{1773725647879019262913712293886257344128658191130053474400713931727} a^{17} - \frac{875241438996746698457995530676084913500465935931621083853833594497}{1773725647879019262913712293886257344128658191130053474400713931727} a^{15} + \frac{60466585409600690161884940554067928522232302136838937931661601715}{1773725647879019262913712293886257344128658191130053474400713931727} a^{13} - \frac{485726025494363276618532225177901328507276159276176678054676853964}{1773725647879019262913712293886257344128658191130053474400713931727} a^{11} - \frac{165959056844894622769943648680324149238471581618850397529733554640}{1773725647879019262913712293886257344128658191130053474400713931727} a^{9} + \frac{859689877861723093023853377449202712707409616825890819515600280056}{1773725647879019262913712293886257344128658191130053474400713931727} a^{7} + \frac{396192289210241788563947865445458855817839968817628718819661083791}{1773725647879019262913712293886257344128658191130053474400713931727} a^{5} + \frac{752861951381189501111726070124087350492556649186302928822363945813}{1773725647879019262913712293886257344128658191130053474400713931727} a^{3} - \frac{82225258859283191951777519920807440004636171881109873738133652257}{1773725647879019262913712293886257344128658191130053474400713931727} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $21$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{44}$ (as 44T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.0.11279504.1, 11.11.31181719929966183601.1, 22.22.86534669543385676516186776267386878120889.1

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 $44$ $22^{2}$ $44$ $22^{2}$ $44$ $22^{2}$ $44$ $44$ $44$ $44$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{11}$ $44$ $44$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{4}$ $22^{2}$ $44$

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
89Data not computed