Properties

Label 36.0.459...733.1
Degree $36$
Signature $[0, 18]$
Discriminant $4.591\times 10^{59}$
Root discriminant \(45.42\)
Ramified primes $3,13$
Class number $729$ (GRH)
Class group [3, 3, 9, 9] (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 3*x^34 - x^33 + 9*x^32 - 6*x^31 + 28*x^30 - 27*x^29 + 90*x^28 - 109*x^27 + 297*x^26 - 417*x^25 + 1000*x^24 + 1845*x^23 + 3417*x^22 + 4535*x^21 + 8406*x^20 + 10188*x^19 + 20683*x^18 + 22158*x^17 + 51861*x^16 + 45791*x^15 + 133425*x^14 + 85512*x^13 + 354484*x^12 + 123111*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1)
 
gp: K = bnfinit(y^36 + 3*y^34 - y^33 + 9*y^32 - 6*y^31 + 28*y^30 - 27*y^29 + 90*y^28 - 109*y^27 + 297*y^26 - 417*y^25 + 1000*y^24 + 1845*y^23 + 3417*y^22 + 4535*y^21 + 8406*y^20 + 10188*y^19 + 20683*y^18 + 22158*y^17 + 51861*y^16 + 45791*y^15 + 133425*y^14 + 85512*y^13 + 354484*y^12 + 123111*y^11 + 42756*y^10 + 14849*y^9 + 5157*y^8 + 1791*y^7 + 622*y^6 + 216*y^5 + 75*y^4 + 26*y^3 + 9*y^2 + 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 3*x^34 - x^33 + 9*x^32 - 6*x^31 + 28*x^30 - 27*x^29 + 90*x^28 - 109*x^27 + 297*x^26 - 417*x^25 + 1000*x^24 + 1845*x^23 + 3417*x^22 + 4535*x^21 + 8406*x^20 + 10188*x^19 + 20683*x^18 + 22158*x^17 + 51861*x^16 + 45791*x^15 + 133425*x^14 + 85512*x^13 + 354484*x^12 + 123111*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 3*x^34 - x^33 + 9*x^32 - 6*x^31 + 28*x^30 - 27*x^29 + 90*x^28 - 109*x^27 + 297*x^26 - 417*x^25 + 1000*x^24 + 1845*x^23 + 3417*x^22 + 4535*x^21 + 8406*x^20 + 10188*x^19 + 20683*x^18 + 22158*x^17 + 51861*x^16 + 45791*x^15 + 133425*x^14 + 85512*x^13 + 354484*x^12 + 123111*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1)
 

\( x^{36} + 3 x^{34} - x^{33} + 9 x^{32} - 6 x^{31} + 28 x^{30} - 27 x^{29} + 90 x^{28} - 109 x^{27} + 297 x^{26} - 417 x^{25} + 1000 x^{24} + 1845 x^{23} + 3417 x^{22} + 4535 x^{21} + 8406 x^{20} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(459146050215773460843525344476713987772454059613596693862733\) \(\medspace = 3^{48}\cdot 13^{33}\) 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:  \(45.42\)
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:  $3^{4/3}13^{11/12}\approx 45.423062587011394$
Ramified primes:   \(3\), \(13\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(117=3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{117}(1,·)$, $\chi_{117}(4,·)$, $\chi_{117}(7,·)$, $\chi_{117}(10,·)$, $\chi_{117}(16,·)$, $\chi_{117}(19,·)$, $\chi_{117}(22,·)$, $\chi_{117}(25,·)$, $\chi_{117}(28,·)$, $\chi_{117}(31,·)$, $\chi_{117}(34,·)$, $\chi_{117}(37,·)$, $\chi_{117}(40,·)$, $\chi_{117}(43,·)$, $\chi_{117}(46,·)$, $\chi_{117}(49,·)$, $\chi_{117}(55,·)$, $\chi_{117}(58,·)$, $\chi_{117}(61,·)$, $\chi_{117}(64,·)$, $\chi_{117}(67,·)$, $\chi_{117}(70,·)$, $\chi_{117}(73,·)$, $\chi_{117}(76,·)$, $\chi_{117}(79,·)$, $\chi_{117}(82,·)$, $\chi_{117}(85,·)$, $\chi_{117}(88,·)$, $\chi_{117}(94,·)$, $\chi_{117}(97,·)$, $\chi_{117}(100,·)$, $\chi_{117}(103,·)$, $\chi_{117}(106,·)$, $\chi_{117}(109,·)$, $\chi_{117}(112,·)$, $\chi_{117}(115,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{405821287}a^{25}+\frac{179067424}{405821287}a^{24}+\frac{3652395}{405821287}a^{23}+\frac{131380984}{405821287}a^{22}-\frac{168110239}{405821287}a^{21}-\frac{15330730}{405821287}a^{20}+\frac{175930873}{405821287}a^{19}+\frac{122118049}{405821287}a^{18}+\frac{137302062}{405821287}a^{17}+\frac{190423274}{405821287}a^{16}-\frac{116033150}{405821287}a^{15}+\frac{28146473}{405821287}a^{14}-\frac{132701437}{405821287}a^{13}+\frac{100237981}{405821287}a^{12}+\frac{20436748}{405821287}a^{11}-\frac{132703546}{405821287}a^{10}-\frac{38927737}{405821287}a^{9}-\frac{12726099}{405821287}a^{8}+\frac{15920335}{405821287}a^{7}+\frac{749440}{405821287}a^{6}+\frac{60487104}{405821287}a^{5}-\frac{13672015}{405821287}a^{4}+\frac{180711872}{405821287}a^{3}-\frac{101503149}{405821287}a^{2}+\frac{149986344}{405821287}a-\frac{79400032}{405821287}$, $\frac{1}{405821287}a^{26}-\frac{100234588}{405821287}a^{13}-\frac{145640894}{405821287}$, $\frac{1}{405821287}a^{27}-\frac{100234588}{405821287}a^{14}-\frac{145640894}{405821287}a$, $\frac{1}{405821287}a^{28}-\frac{100234588}{405821287}a^{15}-\frac{145640894}{405821287}a^{2}$, $\frac{1}{405821287}a^{29}-\frac{100234588}{405821287}a^{16}-\frac{145640894}{405821287}a^{3}$, $\frac{1}{405821287}a^{30}-\frac{100234588}{405821287}a^{17}-\frac{145640894}{405821287}a^{4}$, $\frac{1}{405821287}a^{31}-\frac{100234588}{405821287}a^{18}-\frac{145640894}{405821287}a^{5}$, $\frac{1}{405821287}a^{32}-\frac{100234588}{405821287}a^{19}-\frac{145640894}{405821287}a^{6}$, $\frac{1}{405821287}a^{33}-\frac{100234588}{405821287}a^{20}-\frac{145640894}{405821287}a^{7}$, $\frac{1}{405821287}a^{34}-\frac{100234588}{405821287}a^{21}-\frac{145640894}{405821287}a^{8}$, $\frac{1}{405821287}a^{35}-\frac{100234588}{405821287}a^{22}-\frac{145640894}{405821287}a^{9}$ 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

$C_{3}\times C_{3}\times C_{9}\times C_{9}$, which has order $729$ (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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{10251}{405821287} a^{29} + \frac{34737096}{405821287} a^{16} - \frac{9688000409}{405821287} a^{3} \)  (order $26$) 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:   $\frac{29753}{405821287}a^{30}+\frac{3417}{405821287}a^{28}+\frac{100941399}{405821287}a^{17}+\frac{11579032}{405821287}a^{15}-\frac{27895485399}{405821287}a^{4}-\frac{3364607232}{405821287}a^{2}$, $\frac{29753}{405821287}a^{30}+\frac{100941399}{405821287}a^{17}-\frac{27895485399}{405821287}a^{4}-1$, $\frac{50998407}{405821287}a^{34}-\frac{17711541}{405821287}a^{33}+\frac{152995221}{405821287}a^{32}-\frac{101996814}{405821287}a^{31}+\frac{475985132}{405821287}a^{30}-\frac{458985663}{405821287}a^{29}+\frac{1529952210}{405821287}a^{28}-\frac{1852942121}{405821287}a^{27}+\frac{5048842293}{405821287}a^{26}-\frac{7088778573}{405821287}a^{25}+\frac{16999469000}{405821287}a^{24}-\frac{26315176761}{405821287}a^{23}+\frac{58087185573}{405821287}a^{22}+\frac{77092591915}{405821287}a^{21}+\frac{140481482616}{405821287}a^{20}+\frac{173190590172}{405821287}a^{19}+\frac{351600017327}{405821287}a^{18}+\frac{376674234102}{405821287}a^{17}+\frac{881609461809}{405821287}a^{16}+\frac{778422684979}{405821287}a^{15}+\frac{2268154151325}{405821287}a^{14}+\frac{1453658593128}{405821287}a^{13}+\frac{6026039768996}{405821287}a^{12}+\frac{2092821628059}{405821287}a^{11}+\frac{16624466021955}{405821287}a^{10}+\frac{252425115181}{405821287}a^{9}+\frac{87666261633}{405821287}a^{8}+\frac{696383203038}{405821287}a^{7}+\frac{10573669718}{405821287}a^{6}+\frac{3671885304}{405821287}a^{5}+\frac{1274960175}{405821287}a^{4}+\frac{441986194}{405821287}a^{3}+\frac{152995221}{405821287}a^{2}+\frac{50998407}{405821287}a+\frac{422820756}{405821287}$, $\frac{5903847}{405821287}a^{35}-\frac{48948033}{405821287}a^{34}+\frac{17711541}{405821287}a^{33}-\frac{152995221}{405821287}a^{32}+\frac{101996814}{405821287}a^{31}-\frac{475955379}{405821287}a^{30}+\frac{458995914}{405821287}a^{29}-\frac{1529948793}{405821287}a^{28}+\frac{1852943121}{405821287}a^{27}-\frac{5048842293}{405821287}a^{26}+\frac{7088778573}{405821287}a^{25}-\frac{16999469000}{405821287}a^{24}+\frac{26315176761}{405821287}a^{23}-\frac{38055435285}{405821287}a^{22}-\frac{70135675686}{405821287}a^{21}-\frac{140481482616}{405821287}a^{20}-\frac{173190590172}{405821287}a^{19}-\frac{351600017327}{405821287}a^{18}-\frac{376573292703}{405821287}a^{17}-\frac{881574724713}{405821287}a^{16}-\frac{778411105947}{405821287}a^{15}-\frac{2268150881436}{405821287}a^{14}-\frac{1453658593128}{405821287}a^{13}-\frac{6026039768996}{405821287}a^{12}-\frac{2092821628059}{405821287}a^{11}-\frac{16624466021955}{405821287}a^{10}-\frac{5773616408331}{405821287}a^{9}-\frac{2005155874845}{405821287}a^{8}-\frac{696383203038}{405821287}a^{7}-\frac{10573669718}{405821287}a^{6}-\frac{3671885304}{405821287}a^{5}-\frac{29170445574}{405821287}a^{4}-\frac{10129986603}{405821287}a^{3}-\frac{3517602453}{405821287}a^{2}-\frac{1219514235}{405821287}a-\frac{422820756}{405821287}$, $\frac{140940252}{405821287}a^{35}-\frac{48948033}{405821287}a^{34}+\frac{439820225}{405821287}a^{33}-\frac{293688198}{405821287}a^{32}+\frac{1370459082}{405821287}a^{31}-\frac{1321596891}{405821287}a^{30}+\frac{4405312719}{405821287}a^{29}-\frac{5335335597}{405821287}a^{28}+\frac{14537565801}{405821287}a^{27}-\frac{20411329761}{405821287}a^{26}+\frac{48948033000}{405821287}a^{25}-\frac{75771554084}{405821287}a^{24}+\frac{167255428761}{405821287}a^{23}+\frac{201947579367}{405821287}a^{22}+\frac{411457165398}{405821287}a^{21}+\frac{496266506406}{405821287}a^{20}+\frac{1012392166539}{405821287}a^{19}+\frac{1084299270049}{405821287}a^{18}+\frac{2538493939413}{405821287}a^{17}+\frac{2241344642007}{405821287}a^{16}+\frac{6530891303025}{405821287}a^{15}+\frac{4185644197896}{405821287}a^{14}+\frac{17351294529972}{405821287}a^{13}+\frac{6026041290663}{405821287}a^{12}+\frac{47868242661909}{405821287}a^{11}+\frac{726829342017}{405821287}a^{10}+\frac{5773616299331}{405821287}a^{9}+\frac{87665927103}{405821287}a^{8}+\frac{696382830585}{405821287}a^{7}+\frac{10572775128}{405821287}a^{6}+\frac{83992951440}{405821287}a^{5}+\frac{1272648858}{405821287}a^{4}+\frac{10128532706}{405821287}a^{3}+\frac{146844099}{405821287}a^{2}+\frac{48948033}{405821287}a+1$, $\frac{2417}{405821287}a^{28}+\frac{8309143}{405821287}a^{15}-\frac{1790270117}{405821287}a^{2}$, $\frac{1091027}{405821287}a^{33}+\frac{3701864032}{405821287}a^{20}-\frac{1020274912667}{405821287}a^{7}$, $\frac{5903847}{405821287}a^{35}-\frac{50998407}{405821287}a^{34}+\frac{18337771}{405821287}a^{33}-\frac{152995221}{405821287}a^{32}+\frac{102082656}{405821287}a^{31}-\frac{475955379}{405821287}a^{30}+\frac{458985663}{405821287}a^{29}-\frac{1529952210}{405821287}a^{28}+\frac{1852942121}{405821287}a^{27}-\frac{5048842293}{405821287}a^{26}+\frac{7088778573}{405821287}a^{25}-\frac{16999469000}{405821287}a^{24}+\frac{26315176761}{405821287}a^{23}-\frac{38055435285}{405821287}a^{22}-\frac{77092591915}{405821287}a^{21}-\frac{138356673983}{405821287}a^{20}-\frac{173190590172}{405821287}a^{19}-\frac{351308772162}{405821287}a^{18}-\frac{376573292703}{405821287}a^{17}-\frac{881609461809}{405821287}a^{16}-\frac{778422684979}{405821287}a^{15}-\frac{2268154151325}{405821287}a^{14}-\frac{1453658593128}{405821287}a^{13}-\frac{6026039768996}{405821287}a^{12}-\frac{2092821628059}{405821287}a^{11}-\frac{16624466021955}{405821287}a^{10}-\frac{5773616408331}{405821287}a^{9}-\frac{87666261633}{405821287}a^{8}-\frac{1281998508132}{405821287}a^{7}-\frac{10573669718}{405821287}a^{6}-\frac{83993734269}{405821287}a^{5}-\frac{29170445574}{405821287}a^{4}-\frac{441986194}{405821287}a^{3}-\frac{152995221}{405821287}a^{2}-\frac{50998407}{405821287}a-\frac{16999469}{405821287}$, $\frac{54851880}{405821287}a^{35}+\frac{712072}{405821287}a^{34}+\frac{147556171}{405821287}a^{33}-\frac{48700758}{405821287}a^{32}+\frac{440562050}{405821287}a^{31}-\frac{293658445}{405821287}a^{30}+\frac{1370544924}{405821287}a^{29}-\frac{1321596891}{405821287}a^{28}+\frac{4405322970}{405821287}a^{27}-\frac{5335336597}{405821287}a^{26}+\frac{14537564801}{405821287}a^{25}-\frac{20411329761}{405821287}a^{24}+\frac{48948033000}{405821287}a^{23}+\frac{110340871173}{405821287}a^{22}+\frac{169671482559}{405821287}a^{21}+\frac{224395383453}{405821287}a^{20}+\frac{412296163797}{405821287}a^{19}+\frac{498783501603}{405821287}a^{18}+\frac{1012493107938}{405821287}a^{17}+\frac{1084590515214}{405821287}a^{16}+\frac{2538493939413}{405821287}a^{15}+\frac{2241379379103}{405821287}a^{14}+\frac{6530888033136}{405821287}a^{13}+\frac{4185640928007}{405821287}a^{12}+\frac{17351294529972}{405821287}a^{11}+\frac{6026041290663}{405821287}a^{10}-\frac{3428369194202}{405821287}a^{9}+\frac{60892187958}{405821287}a^{8}-\frac{413512147878}{405821287}a^{7}-\frac{143611619383}{405821287}a^{6}+\frac{2550191127}{405821287}a^{5}-\frac{17322710271}{405821287}a^{4}+\frac{3671102475}{405821287}a^{3}+\frac{1272648858}{405821287}a^{2}+\frac{34711010}{405821287}a+\frac{503717353}{405821287}$, $\frac{5903847}{405821287}a^{35}+\frac{2050374}{405821287}a^{34}+\frac{5834}{405821287}a^{28}+\frac{20031750288}{405821287}a^{22}+\frac{6956916229}{405821287}a^{21}+\frac{19888175}{405821287}a^{15}-\frac{5521191293150}{405821287}a^{9}-\frac{1917489613212}{405821287}a^{8}-\frac{5154877349}{405821287}a^{2}$, $\frac{3853473}{405821287}a^{34}-\frac{10251}{405821287}a^{29}+\frac{13074834059}{405821287}a^{21}-\frac{34737096}{405821287}a^{16}-\frac{3603701679938}{405821287}a^{8}+\frac{9688000409}{405821287}a^{3}+1$, $\frac{69088903}{405821287}a^{35}+\frac{48948033}{405821287}a^{34}+\frac{207266709}{405821287}a^{33}+\frac{77755196}{405821287}a^{32}+\frac{572852094}{405821287}a^{31}+\frac{25998879}{405821287}a^{30}+\frac{1640801086}{405821287}a^{29}-\frac{494855457}{405821287}a^{28}+\frac{4896404379}{405821287}a^{27}-\frac{3125367457}{405821287}a^{26}+\frac{15184070011}{405821287}a^{25}-\frac{14272507750}{405821287}a^{24}+\frac{48677573239}{405821287}a^{23}+\frac{176417059035}{405821287}a^{22}+\frac{326385902436}{405821287}a^{21}+\frac{480573603866}{405821287}a^{20}+\frac{802740648273}{405821287}a^{19}+\frac{1115334909162}{405821287}a^{18}+\frac{1927648340953}{405821287}a^{17}+\frac{2543264079213}{405821287}a^{16}+\frac{4667610113697}{405821287}a^{15}+\frac{5702143896686}{405821287}a^{14}+\frac{11459566261878}{405821287}a^{13}+\frac{12438826615615}{405821287}a^{12}+\frac{28676551619059}{405821287}a^{11}+\frac{25856898467205}{405821287}a^{10}+\frac{8980006427331}{405821287}a^{9}+\frac{3118723219595}{405821287}a^{8}+\frac{1083120814788}{405821287}a^{7}+\frac{376163231454}{405821287}a^{6}+\frac{130639224769}{405821287}a^{5}+\frac{45368879574}{405821287}a^{4}+\frac{15754442853}{405821287}a^{3}+\frac{5467413953}{405821287}a^{2}+\frac{1894448985}{405821287}a+\frac{241977719}{405821287}$, $\frac{5903847}{405821287}a^{35}-\frac{712072}{405821287}a^{34}+\frac{712072}{405821287}a^{33}+\frac{20031750288}{405821287}a^{22}-\frac{2416053798}{405821287}a^{21}+\frac{2416053798}{405821287}a^{20}-\frac{5521191293150}{405821287}a^{9}+\frac{665937154059}{405821287}a^{8}-\frac{665937154059}{405821287}a^{7}$, $\frac{140940252}{405821287}a^{35}+\frac{46897659}{405821287}a^{34}+\frac{408583733}{405821287}a^{33}-\frac{6151122}{405821287}a^{32}+\frac{1178853540}{405821287}a^{31}-\frac{427037099}{405821287}a^{30}+\frac{3542711742}{405821287}a^{29}-\frac{2459964837}{405821287}a^{28}+\frac{11055172325}{405821287}a^{27}-\frac{10922606253}{405821287}a^{26}+\frac{35625481812}{405821287}a^{25}-\frac{43822990084}{405821287}a^{24}+\frac{117799051438}{405821287}a^{23}+\frac{311115822555}{405821287}a^{22}+\frac{556343903138}{405821287}a^{21}+\frac{767242189188}{405821287}a^{20}+\frac{1337884136571}{405821287}a^{19}+\frac{1745382664426}{405821287}a^{18}+\frac{3246410220525}{405821287}a^{17}+\frac{3898263856707}{405821287}a^{16}+\frac{7993847997149}{405821287}a^{15}+\frac{8448381349596}{405821287}a^{14}+\frac{20083280134740}{405821287}a^{13}+\frac{17351296051639}{405821287}a^{12}+\frac{51801462324513}{405821287}a^{11}+\frac{31970605981971}{405821287}a^{10}+\frac{6248020526167}{405821287}a^{9}+\frac{252424671651}{405821287}a^{8}+\frac{87665554650}{405821287}a^{7}+\frac{30444781936}{405821287}a^{6}+\frac{10571992299}{405821287}a^{5}+\frac{3668791158}{405821287}a^{4}+\frac{1271194961}{405821287}a^{3}+\frac{434381175}{405821287}a^{2}+\frac{144793725}{405821287}a+\frac{31948564}{405821287}$, $\frac{2050374}{405821287}a^{34}+\frac{712072}{405821287}a^{33}-\frac{2417}{405821287}a^{27}+\frac{6956916229}{405821287}a^{21}+\frac{2416053798}{405821287}a^{20}-\frac{8309143}{405821287}a^{14}-\frac{1917489613212}{405821287}a^{8}-\frac{665937154059}{405821287}a^{7}+\frac{2196091404}{405821287}a$, $\frac{224978250}{405821287}a^{35}-\frac{77043124}{405821287}a^{34}+\frac{674934750}{405821287}a^{33}-\frac{449956500}{405821287}a^{32}+\frac{2099797000}{405821287}a^{31}-\frac{2024804250}{405821287}a^{30}+\frac{6749347500}{405821287}a^{29}-\frac{8174209750}{405821287}a^{28}+\frac{22272845750}{405821287}a^{27}-\frac{31271976750}{405821287}a^{26}+\frac{74992750000}{405821287}a^{25}-\frac{116088778834}{405821287}a^{24}+\frac{256250226750}{405821287}a^{23}+\frac{340092121250}{405821287}a^{22}+\frac{623432140271}{405821287}a^{21}+\frac{764026137000}{405821287}a^{20}+\frac{1551075048250}{405821287}a^{19}+\frac{1661689354500}{405821287}a^{18}+\frac{3889199007750}{405821287}a^{17}+\frac{3433993015250}{405821287}a^{16}+\frac{10005907668750}{405821287}a^{15}+\frac{6412776768111}{405821287}a^{14}+\frac{26583729991000}{405821287}a^{13}+\frac{9232432445250}{405821287}a^{12}+\frac{73338403126381}{405821287}a^{11}+\frac{1113567344750}{405821287}a^{10}+\frac{386737611750}{405821287}a^{9}+\frac{2051801628462}{405821287}a^{8}+\frac{46645490500}{405821287}a^{7}+\frac{16198434000}{405821287}a^{6}+\frac{5624456250}{405821287}a^{5}+\frac{1949811500}{405821287}a^{4}+\frac{674934750}{405821287}a^{3}+\frac{224978250}{405821287}a^{2}+\frac{1243508578}{405821287}a$, $\frac{85842}{405821287}a^{31}+\frac{3417}{405821287}a^{29}+\frac{1000}{405821287}a^{27}+\frac{291245165}{405821287}a^{18}+\frac{11579032}{405821287}a^{16}+\frac{3269889}{405821287}a^{14}-\frac{80321848965}{405821287}a^{5}-\frac{3364607232}{405821287}a^{3}-\frac{1168515828}{405821287}a$ 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:  \( 20980577392492.816 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 20980577392492.816 \cdot 729}{26\cdot\sqrt{459146050215773460843525344476713987772454059613596693862733}}\cr\approx \mathstrut & 0.202224523334538 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 + 3*x^34 - x^33 + 9*x^32 - 6*x^31 + 28*x^30 - 27*x^29 + 90*x^28 - 109*x^27 + 297*x^26 - 417*x^25 + 1000*x^24 + 1845*x^23 + 3417*x^22 + 4535*x^21 + 8406*x^20 + 10188*x^19 + 20683*x^18 + 22158*x^17 + 51861*x^16 + 45791*x^15 + 133425*x^14 + 85512*x^13 + 354484*x^12 + 123111*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 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^36 + 3*x^34 - x^33 + 9*x^32 - 6*x^31 + 28*x^30 - 27*x^29 + 90*x^28 - 109*x^27 + 297*x^26 - 417*x^25 + 1000*x^24 + 1845*x^23 + 3417*x^22 + 4535*x^21 + 8406*x^20 + 10188*x^19 + 20683*x^18 + 22158*x^17 + 51861*x^16 + 45791*x^15 + 133425*x^14 + 85512*x^13 + 354484*x^12 + 123111*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 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^36 + 3*x^34 - x^33 + 9*x^32 - 6*x^31 + 28*x^30 - 27*x^29 + 90*x^28 - 109*x^27 + 297*x^26 - 417*x^25 + 1000*x^24 + 1845*x^23 + 3417*x^22 + 4535*x^21 + 8406*x^20 + 10188*x^19 + 20683*x^18 + 22158*x^17 + 51861*x^16 + 45791*x^15 + 133425*x^14 + 85512*x^13 + 354484*x^12 + 123111*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 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^36 + 3*x^34 - x^33 + 9*x^32 - 6*x^31 + 28*x^30 - 27*x^29 + 90*x^28 - 109*x^27 + 297*x^26 - 417*x^25 + 1000*x^24 + 1845*x^23 + 3417*x^22 + 4535*x^21 + 8406*x^20 + 10188*x^19 + 20683*x^18 + 22158*x^17 + 51861*x^16 + 45791*x^15 + 133425*x^14 + 85512*x^13 + 354484*x^12 + 123111*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 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

$C_3\times C_{12}$ (as 36T3):

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 36
The 36 conjugacy class representatives for $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.2, 3.3.13689.1, 4.0.2197.1, 6.6.14414517.1, \(\Q(\zeta_{13})^+\), 6.6.2436053373.2, 6.6.2436053373.1, 9.9.2565164201769.1, 12.0.456488925854205933.1, \(\Q(\zeta_{13})\), 12.0.77146628469360802677.1, 12.0.77146628469360802677.2, 18.18.14456408038335708501176406117.1

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 ${\href{/padicField/2.12.0.1}{12} }^{3}$ R ${\href{/padicField/5.12.0.1}{12} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }^{3}$ ${\href{/padicField/11.12.0.1}{12} }^{3}$ R ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.12.0.1}{12} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{12}$ ${\href{/padicField/31.12.0.1}{12} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }^{3}$ ${\href{/padicField/41.12.0.1}{12} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }^{6}$ ${\href{/padicField/47.12.0.1}{12} }^{3}$ ${\href{/padicField/53.1.0.1}{1} }^{36}$ ${\href{/padicField/59.12.0.1}{12} }^{3}$

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
\(3\) Copy content Toggle raw display 3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
\(13\) Copy content Toggle raw display Deg $36$$12$$3$$33$