Properties

Label 36.0.392...656.1
Degree $36$
Signature $[0, 18]$
Discriminant $3.929\times 10^{62}$
Root discriminant \(54.79\)
Ramified primes $2,19$
Class number $3249$ (GRH)
Class group [19, 171] (GRH)
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 49*x^32 + 932*x^28 + 8695*x^24 + 41461*x^20 + 96055*x^16 + 93536*x^12 + 28314*x^8 + 1365*x^4 + 1)
 
gp: K = bnfinit(y^36 + 49*y^32 + 932*y^28 + 8695*y^24 + 41461*y^20 + 96055*y^16 + 93536*y^12 + 28314*y^8 + 1365*y^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 49*x^32 + 932*x^28 + 8695*x^24 + 41461*x^20 + 96055*x^16 + 93536*x^12 + 28314*x^8 + 1365*x^4 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 49*x^32 + 932*x^28 + 8695*x^24 + 41461*x^20 + 96055*x^16 + 93536*x^12 + 28314*x^8 + 1365*x^4 + 1)
 

\( x^{36} + 49 x^{32} + 932 x^{28} + 8695 x^{24} + 41461 x^{20} + 96055 x^{16} + 93536 x^{12} + 28314 x^{8} + 1365 x^{4} + 1 \) 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:   \(392893567271872510941083606170645734076396278452324169894854656\) \(\medspace = 2^{72}\cdot 19^{32}\) 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:  \(54.79\)
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:  $2^{2}19^{8/9}\approx 54.79360300815596$
Ramified primes:   \(2\), \(19\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(152=2^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{152}(1,·)$, $\chi_{152}(131,·)$, $\chi_{152}(5,·)$, $\chi_{152}(7,·)$, $\chi_{152}(9,·)$, $\chi_{152}(11,·)$, $\chi_{152}(17,·)$, $\chi_{152}(149,·)$, $\chi_{152}(23,·)$, $\chi_{152}(25,·)$, $\chi_{152}(35,·)$, $\chi_{152}(39,·)$, $\chi_{152}(43,·)$, $\chi_{152}(45,·)$, $\chi_{152}(47,·)$, $\chi_{152}(49,·)$, $\chi_{152}(137,·)$, $\chi_{152}(61,·)$, $\chi_{152}(63,·)$, $\chi_{152}(139,·)$, $\chi_{152}(73,·)$, $\chi_{152}(55,·)$, $\chi_{152}(77,·)$, $\chi_{152}(81,·)$, $\chi_{152}(83,·)$, $\chi_{152}(85,·)$, $\chi_{152}(87,·)$, $\chi_{152}(93,·)$, $\chi_{152}(99,·)$, $\chi_{152}(101,·)$, $\chi_{152}(111,·)$, $\chi_{152}(115,·)$, $\chi_{152}(119,·)$, $\chi_{152}(121,·)$, $\chi_{152}(123,·)$, $\chi_{152}(125,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{37}a^{28}+\frac{16}{37}a^{24}+\frac{11}{37}a^{16}-\frac{9}{37}a^{12}+\frac{10}{37}a^{4}+\frac{12}{37}$, $\frac{1}{37}a^{29}+\frac{16}{37}a^{25}+\frac{11}{37}a^{17}-\frac{9}{37}a^{13}+\frac{10}{37}a^{5}+\frac{12}{37}a$, $\frac{1}{37}a^{30}+\frac{16}{37}a^{26}+\frac{11}{37}a^{18}-\frac{9}{37}a^{14}+\frac{10}{37}a^{6}+\frac{12}{37}a^{2}$, $\frac{1}{37}a^{31}+\frac{16}{37}a^{27}+\frac{11}{37}a^{19}-\frac{9}{37}a^{15}+\frac{10}{37}a^{7}+\frac{12}{37}a^{3}$, $\frac{1}{27\!\cdots\!89}a^{32}-\frac{11925490864898}{27\!\cdots\!89}a^{28}+\frac{64\!\cdots\!47}{27\!\cdots\!89}a^{24}-\frac{55\!\cdots\!76}{27\!\cdots\!89}a^{20}+\frac{11\!\cdots\!87}{27\!\cdots\!89}a^{16}-\frac{85\!\cdots\!15}{27\!\cdots\!89}a^{12}+\frac{10\!\cdots\!72}{27\!\cdots\!89}a^{8}-\frac{50\!\cdots\!56}{27\!\cdots\!89}a^{4}+\frac{12\!\cdots\!05}{27\!\cdots\!89}$, $\frac{1}{27\!\cdots\!89}a^{33}-\frac{11925490864898}{27\!\cdots\!89}a^{29}+\frac{64\!\cdots\!47}{27\!\cdots\!89}a^{25}-\frac{55\!\cdots\!76}{27\!\cdots\!89}a^{21}+\frac{11\!\cdots\!87}{27\!\cdots\!89}a^{17}-\frac{85\!\cdots\!15}{27\!\cdots\!89}a^{13}+\frac{10\!\cdots\!72}{27\!\cdots\!89}a^{9}-\frac{50\!\cdots\!56}{27\!\cdots\!89}a^{5}+\frac{12\!\cdots\!05}{27\!\cdots\!89}a$, $\frac{1}{27\!\cdots\!89}a^{34}-\frac{11925490864898}{27\!\cdots\!89}a^{30}+\frac{64\!\cdots\!47}{27\!\cdots\!89}a^{26}-\frac{55\!\cdots\!76}{27\!\cdots\!89}a^{22}+\frac{11\!\cdots\!87}{27\!\cdots\!89}a^{18}-\frac{85\!\cdots\!15}{27\!\cdots\!89}a^{14}+\frac{10\!\cdots\!72}{27\!\cdots\!89}a^{10}-\frac{50\!\cdots\!56}{27\!\cdots\!89}a^{6}+\frac{12\!\cdots\!05}{27\!\cdots\!89}a^{2}$, $\frac{1}{27\!\cdots\!89}a^{35}-\frac{11925490864898}{27\!\cdots\!89}a^{31}+\frac{64\!\cdots\!47}{27\!\cdots\!89}a^{27}-\frac{55\!\cdots\!76}{27\!\cdots\!89}a^{23}+\frac{11\!\cdots\!87}{27\!\cdots\!89}a^{19}-\frac{85\!\cdots\!15}{27\!\cdots\!89}a^{15}+\frac{10\!\cdots\!72}{27\!\cdots\!89}a^{11}-\frac{50\!\cdots\!56}{27\!\cdots\!89}a^{7}+\frac{12\!\cdots\!05}{27\!\cdots\!89}a^{3}$ 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

$C_{19}\times C_{171}$, which has order $3249$ (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:   \( \frac{4933715287540570}{27985118605791989} a^{35} + \frac{241728563142432074}{27985118605791989} a^{31} + \frac{4597078388635635768}{27985118605791989} a^{27} + \frac{42877073860924213654}{27985118605791989} a^{23} + \frac{204358106636401389981}{27985118605791989} a^{19} + \frac{472982025320174328996}{27985118605791989} a^{15} + \frac{459425171055352514653}{27985118605791989} a^{11} + \frac{137874115120972430578}{27985118605791989} a^{7} + \frac{6314209051969526189}{27985118605791989} a^{3} \)  (order $8$) 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:   $\frac{13\!\cdots\!81}{27\!\cdots\!89}a^{34}+\frac{65\!\cdots\!05}{27\!\cdots\!89}a^{30}+\frac{12\!\cdots\!59}{27\!\cdots\!89}a^{26}+\frac{11\!\cdots\!38}{27\!\cdots\!89}a^{22}+\frac{55\!\cdots\!87}{27\!\cdots\!89}a^{18}+\frac{12\!\cdots\!58}{27\!\cdots\!89}a^{14}+\frac{12\!\cdots\!46}{27\!\cdots\!89}a^{10}+\frac{38\!\cdots\!16}{27\!\cdots\!89}a^{6}+\frac{19\!\cdots\!39}{27\!\cdots\!89}a^{2}$, $\frac{81066156645912}{27\!\cdots\!89}a^{33}+\frac{39\!\cdots\!59}{27\!\cdots\!89}a^{29}+\frac{75\!\cdots\!11}{27\!\cdots\!89}a^{25}+\frac{71\!\cdots\!06}{27\!\cdots\!89}a^{21}+\frac{34\!\cdots\!27}{27\!\cdots\!89}a^{17}+\frac{80\!\cdots\!74}{27\!\cdots\!89}a^{13}+\frac{81\!\cdots\!43}{27\!\cdots\!89}a^{9}+\frac{28\!\cdots\!47}{27\!\cdots\!89}a^{5}+\frac{23\!\cdots\!92}{27\!\cdots\!89}a$, $\frac{482482821130913}{27\!\cdots\!89}a^{34}+\frac{23\!\cdots\!58}{27\!\cdots\!89}a^{30}+\frac{45\!\cdots\!91}{27\!\cdots\!89}a^{26}+\frac{42\!\cdots\!42}{27\!\cdots\!89}a^{22}+\frac{20\!\cdots\!22}{27\!\cdots\!89}a^{18}+\frac{46\!\cdots\!31}{27\!\cdots\!89}a^{14}+\frac{46\!\cdots\!39}{27\!\cdots\!89}a^{10}+\frac{14\!\cdots\!70}{27\!\cdots\!89}a^{6}+\frac{86\!\cdots\!88}{27\!\cdots\!89}a^{2}$, $\frac{78992634535392}{27\!\cdots\!89}a^{32}+\frac{38\!\cdots\!96}{27\!\cdots\!89}a^{28}+\frac{73\!\cdots\!11}{27\!\cdots\!89}a^{24}+\frac{68\!\cdots\!92}{27\!\cdots\!89}a^{20}+\frac{32\!\cdots\!78}{27\!\cdots\!89}a^{16}+\frac{72\!\cdots\!04}{27\!\cdots\!89}a^{12}+\frac{67\!\cdots\!83}{27\!\cdots\!89}a^{8}+\frac{16\!\cdots\!60}{27\!\cdots\!89}a^{4}+\frac{23\!\cdots\!46}{27\!\cdots\!89}$, $\frac{142805734603688}{27\!\cdots\!89}a^{33}+\frac{70\!\cdots\!96}{27\!\cdots\!89}a^{29}+\frac{13\!\cdots\!97}{27\!\cdots\!89}a^{25}+\frac{12\!\cdots\!11}{27\!\cdots\!89}a^{21}+\frac{59\!\cdots\!01}{27\!\cdots\!89}a^{17}+\frac{14\!\cdots\!56}{27\!\cdots\!89}a^{13}+\frac{14\!\cdots\!75}{27\!\cdots\!89}a^{9}+\frac{46\!\cdots\!68}{27\!\cdots\!89}a^{5}+\frac{27\!\cdots\!98}{27\!\cdots\!89}a$, $\frac{12\!\cdots\!94}{27\!\cdots\!89}a^{35}+\frac{59\!\cdots\!46}{27\!\cdots\!89}a^{31}+\frac{11\!\cdots\!83}{27\!\cdots\!89}a^{27}+\frac{10\!\cdots\!40}{27\!\cdots\!89}a^{23}+\frac{50\!\cdots\!64}{27\!\cdots\!89}a^{19}+\frac{11\!\cdots\!06}{27\!\cdots\!89}a^{15}+\frac{11\!\cdots\!74}{27\!\cdots\!89}a^{11}+\frac{34\!\cdots\!72}{27\!\cdots\!89}a^{7}+\frac{16\!\cdots\!75}{27\!\cdots\!89}a^{3}$, $\frac{230187034012076}{27\!\cdots\!89}a^{33}+\frac{11\!\cdots\!99}{27\!\cdots\!89}a^{29}+\frac{21\!\cdots\!19}{27\!\cdots\!89}a^{25}+\frac{19\!\cdots\!98}{27\!\cdots\!89}a^{21}+\frac{94\!\cdots\!12}{27\!\cdots\!89}a^{17}+\frac{21\!\cdots\!53}{27\!\cdots\!89}a^{13}+\frac{21\!\cdots\!41}{27\!\cdots\!89}a^{9}+\frac{63\!\cdots\!45}{27\!\cdots\!89}a^{5}+\frac{35\!\cdots\!39}{27\!\cdots\!89}a$, $a$, $\frac{674152244105365}{27\!\cdots\!89}a^{34}+\frac{191669422974452}{27\!\cdots\!89}a^{33}+\frac{33\!\cdots\!34}{27\!\cdots\!89}a^{30}+\frac{93\!\cdots\!76}{27\!\cdots\!89}a^{29}+\frac{62\!\cdots\!43}{27\!\cdots\!89}a^{26}+\frac{17\!\cdots\!52}{27\!\cdots\!89}a^{25}+\frac{58\!\cdots\!02}{27\!\cdots\!89}a^{22}+\frac{16\!\cdots\!60}{27\!\cdots\!89}a^{21}+\frac{27\!\cdots\!31}{27\!\cdots\!89}a^{18}+\frac{78\!\cdots\!09}{27\!\cdots\!89}a^{17}+\frac{64\!\cdots\!50}{27\!\cdots\!89}a^{14}+\frac{18\!\cdots\!19}{27\!\cdots\!89}a^{13}+\frac{63\!\cdots\!82}{27\!\cdots\!89}a^{10}+\frac{17\!\cdots\!43}{27\!\cdots\!89}a^{9}+\frac{19\!\cdots\!28}{27\!\cdots\!89}a^{6}+\frac{49\!\cdots\!58}{27\!\cdots\!89}a^{5}+\frac{10\!\cdots\!65}{27\!\cdots\!89}a^{2}+\frac{17\!\cdots\!77}{27\!\cdots\!89}a+1$, $\frac{49\!\cdots\!70}{27\!\cdots\!89}a^{35}-\frac{13\!\cdots\!30}{27\!\cdots\!89}a^{34}+\frac{191669422974452}{27\!\cdots\!89}a^{33}+\frac{24\!\cdots\!74}{27\!\cdots\!89}a^{31}-\frac{66\!\cdots\!68}{27\!\cdots\!89}a^{30}+\frac{93\!\cdots\!76}{27\!\cdots\!89}a^{29}+\frac{45\!\cdots\!68}{27\!\cdots\!89}a^{27}-\frac{12\!\cdots\!86}{27\!\cdots\!89}a^{26}+\frac{17\!\cdots\!52}{27\!\cdots\!89}a^{25}+\frac{42\!\cdots\!54}{27\!\cdots\!89}a^{23}-\frac{11\!\cdots\!04}{27\!\cdots\!89}a^{22}+\frac{16\!\cdots\!60}{27\!\cdots\!89}a^{21}+\frac{20\!\cdots\!81}{27\!\cdots\!89}a^{19}-\frac{55\!\cdots\!62}{27\!\cdots\!89}a^{18}+\frac{78\!\cdots\!09}{27\!\cdots\!89}a^{17}+\frac{47\!\cdots\!96}{27\!\cdots\!89}a^{15}-\frac{12\!\cdots\!00}{27\!\cdots\!89}a^{14}+\frac{18\!\cdots\!19}{27\!\cdots\!89}a^{13}+\frac{45\!\cdots\!53}{27\!\cdots\!89}a^{11}-\frac{12\!\cdots\!64}{27\!\cdots\!89}a^{10}+\frac{17\!\cdots\!43}{27\!\cdots\!89}a^{9}+\frac{13\!\cdots\!78}{27\!\cdots\!89}a^{7}-\frac{38\!\cdots\!56}{27\!\cdots\!89}a^{6}+\frac{49\!\cdots\!58}{27\!\cdots\!89}a^{5}+\frac{63\!\cdots\!89}{27\!\cdots\!89}a^{3}-\frac{20\!\cdots\!41}{27\!\cdots\!89}a^{2}+\frac{17\!\cdots\!77}{27\!\cdots\!89}a$, $\frac{246601049465924}{756354556913297}a^{35}-\frac{674152244105365}{27\!\cdots\!89}a^{34}+\frac{44\!\cdots\!16}{27\!\cdots\!89}a^{31}-\frac{33\!\cdots\!34}{27\!\cdots\!89}a^{30}+\frac{85\!\cdots\!24}{27\!\cdots\!89}a^{27}-\frac{62\!\cdots\!43}{27\!\cdots\!89}a^{26}+\frac{21\!\cdots\!61}{756354556913297}a^{23}-\frac{58\!\cdots\!02}{27\!\cdots\!89}a^{22}+\frac{37\!\cdots\!80}{27\!\cdots\!89}a^{19}-\frac{27\!\cdots\!31}{27\!\cdots\!89}a^{18}+\frac{87\!\cdots\!83}{27\!\cdots\!89}a^{15}-\frac{64\!\cdots\!50}{27\!\cdots\!89}a^{14}+\frac{23\!\cdots\!28}{756354556913297}a^{11}-\frac{63\!\cdots\!82}{27\!\cdots\!89}a^{10}+\frac{25\!\cdots\!85}{27\!\cdots\!89}a^{7}-\frac{19\!\cdots\!28}{27\!\cdots\!89}a^{6}+\frac{11\!\cdots\!40}{27\!\cdots\!89}a^{3}-\frac{10\!\cdots\!65}{27\!\cdots\!89}a^{2}+1$, $\frac{33\!\cdots\!76}{27\!\cdots\!89}a^{35}+\frac{674152244105365}{27\!\cdots\!89}a^{34}+\frac{16\!\cdots\!07}{27\!\cdots\!89}a^{31}+\frac{33\!\cdots\!34}{27\!\cdots\!89}a^{30}+\frac{31\!\cdots\!88}{27\!\cdots\!89}a^{27}+\frac{62\!\cdots\!43}{27\!\cdots\!89}a^{26}+\frac{29\!\cdots\!44}{27\!\cdots\!89}a^{23}+\frac{58\!\cdots\!02}{27\!\cdots\!89}a^{22}+\frac{13\!\cdots\!80}{27\!\cdots\!89}a^{19}+\frac{27\!\cdots\!31}{27\!\cdots\!89}a^{18}+\frac{32\!\cdots\!08}{27\!\cdots\!89}a^{15}+\frac{64\!\cdots\!50}{27\!\cdots\!89}a^{14}+\frac{31\!\cdots\!92}{27\!\cdots\!89}a^{11}+\frac{63\!\cdots\!82}{27\!\cdots\!89}a^{10}+\frac{96\!\cdots\!00}{27\!\cdots\!89}a^{7}+\frac{19\!\cdots\!28}{27\!\cdots\!89}a^{6}+\frac{50\!\cdots\!45}{27\!\cdots\!89}a^{3}+\frac{10\!\cdots\!65}{27\!\cdots\!89}a^{2}-1$, $\frac{82\!\cdots\!36}{27\!\cdots\!89}a^{35}-\frac{674152244105365}{27\!\cdots\!89}a^{34}+\frac{40\!\cdots\!84}{27\!\cdots\!89}a^{31}-\frac{33\!\cdots\!34}{27\!\cdots\!89}a^{30}+\frac{77\!\cdots\!64}{27\!\cdots\!89}a^{27}-\frac{62\!\cdots\!43}{27\!\cdots\!89}a^{26}+\frac{71\!\cdots\!39}{27\!\cdots\!89}a^{23}-\frac{58\!\cdots\!02}{27\!\cdots\!89}a^{22}+\frac{34\!\cdots\!64}{27\!\cdots\!89}a^{19}-\frac{27\!\cdots\!31}{27\!\cdots\!89}a^{18}+\frac{79\!\cdots\!94}{27\!\cdots\!89}a^{15}-\frac{64\!\cdots\!50}{27\!\cdots\!89}a^{14}+\frac{77\!\cdots\!61}{27\!\cdots\!89}a^{11}-\frac{63\!\cdots\!82}{27\!\cdots\!89}a^{10}+\frac{23\!\cdots\!97}{27\!\cdots\!89}a^{7}-\frac{19\!\cdots\!28}{27\!\cdots\!89}a^{6}+\frac{11\!\cdots\!53}{27\!\cdots\!89}a^{3}-\frac{10\!\cdots\!65}{27\!\cdots\!89}a^{2}+1$, $\frac{674152244105365}{27\!\cdots\!89}a^{35}-\frac{674152244105365}{27\!\cdots\!89}a^{34}+\frac{33\!\cdots\!34}{27\!\cdots\!89}a^{31}-\frac{33\!\cdots\!34}{27\!\cdots\!89}a^{30}+\frac{62\!\cdots\!43}{27\!\cdots\!89}a^{27}-\frac{62\!\cdots\!43}{27\!\cdots\!89}a^{26}+\frac{58\!\cdots\!02}{27\!\cdots\!89}a^{23}-\frac{58\!\cdots\!02}{27\!\cdots\!89}a^{22}+\frac{27\!\cdots\!31}{27\!\cdots\!89}a^{19}-\frac{27\!\cdots\!31}{27\!\cdots\!89}a^{18}+\frac{64\!\cdots\!50}{27\!\cdots\!89}a^{15}-\frac{64\!\cdots\!50}{27\!\cdots\!89}a^{14}+\frac{63\!\cdots\!82}{27\!\cdots\!89}a^{11}-\frac{63\!\cdots\!82}{27\!\cdots\!89}a^{10}+\frac{19\!\cdots\!28}{27\!\cdots\!89}a^{7}-\frac{19\!\cdots\!28}{27\!\cdots\!89}a^{6}+\frac{10\!\cdots\!65}{27\!\cdots\!89}a^{3}-\frac{10\!\cdots\!65}{27\!\cdots\!89}a^{2}+1$, $\frac{49\!\cdots\!70}{27\!\cdots\!89}a^{35}-\frac{191669422974452}{27\!\cdots\!89}a^{33}+\frac{76594519447472}{27\!\cdots\!89}a^{32}+\frac{24\!\cdots\!74}{27\!\cdots\!89}a^{31}-\frac{93\!\cdots\!76}{27\!\cdots\!89}a^{29}+\frac{37\!\cdots\!84}{27\!\cdots\!89}a^{28}+\frac{45\!\cdots\!68}{27\!\cdots\!89}a^{27}-\frac{17\!\cdots\!52}{27\!\cdots\!89}a^{25}+\frac{19\!\cdots\!16}{756354556913297}a^{24}+\frac{42\!\cdots\!54}{27\!\cdots\!89}a^{23}-\frac{16\!\cdots\!60}{27\!\cdots\!89}a^{21}+\frac{66\!\cdots\!24}{27\!\cdots\!89}a^{20}+\frac{20\!\cdots\!81}{27\!\cdots\!89}a^{19}-\frac{78\!\cdots\!09}{27\!\cdots\!89}a^{17}+\frac{31\!\cdots\!64}{27\!\cdots\!89}a^{16}+\frac{47\!\cdots\!96}{27\!\cdots\!89}a^{15}-\frac{18\!\cdots\!19}{27\!\cdots\!89}a^{13}+\frac{19\!\cdots\!47}{756354556913297}a^{12}+\frac{45\!\cdots\!53}{27\!\cdots\!89}a^{11}-\frac{17\!\cdots\!43}{27\!\cdots\!89}a^{9}+\frac{69\!\cdots\!90}{27\!\cdots\!89}a^{8}+\frac{13\!\cdots\!78}{27\!\cdots\!89}a^{7}-\frac{49\!\cdots\!58}{27\!\cdots\!89}a^{5}+\frac{18\!\cdots\!47}{27\!\cdots\!89}a^{4}+\frac{63\!\cdots\!89}{27\!\cdots\!89}a^{3}-\frac{17\!\cdots\!77}{27\!\cdots\!89}a+\frac{865082132153286}{756354556913297}$, $\frac{674152244105365}{27\!\cdots\!89}a^{35}-\frac{674152244105365}{27\!\cdots\!89}a^{34}+\frac{33\!\cdots\!34}{27\!\cdots\!89}a^{31}-\frac{33\!\cdots\!34}{27\!\cdots\!89}a^{30}+\frac{62\!\cdots\!43}{27\!\cdots\!89}a^{27}-\frac{62\!\cdots\!43}{27\!\cdots\!89}a^{26}+\frac{58\!\cdots\!02}{27\!\cdots\!89}a^{23}-\frac{58\!\cdots\!02}{27\!\cdots\!89}a^{22}+\frac{27\!\cdots\!31}{27\!\cdots\!89}a^{19}-\frac{27\!\cdots\!31}{27\!\cdots\!89}a^{18}+\frac{64\!\cdots\!50}{27\!\cdots\!89}a^{15}-\frac{64\!\cdots\!50}{27\!\cdots\!89}a^{14}+\frac{63\!\cdots\!82}{27\!\cdots\!89}a^{11}-\frac{63\!\cdots\!82}{27\!\cdots\!89}a^{10}+\frac{19\!\cdots\!28}{27\!\cdots\!89}a^{7}-\frac{19\!\cdots\!28}{27\!\cdots\!89}a^{6}+\frac{10\!\cdots\!65}{27\!\cdots\!89}a^{3}-\frac{10\!\cdots\!76}{27\!\cdots\!89}a^{2}+a$, $\frac{674152244105365}{27\!\cdots\!89}a^{34}-\frac{142805734603688}{27\!\cdots\!89}a^{33}+\frac{33\!\cdots\!34}{27\!\cdots\!89}a^{30}-\frac{70\!\cdots\!96}{27\!\cdots\!89}a^{29}+\frac{62\!\cdots\!43}{27\!\cdots\!89}a^{26}-\frac{13\!\cdots\!97}{27\!\cdots\!89}a^{25}+\frac{58\!\cdots\!02}{27\!\cdots\!89}a^{22}-\frac{12\!\cdots\!11}{27\!\cdots\!89}a^{21}+\frac{27\!\cdots\!31}{27\!\cdots\!89}a^{18}-\frac{59\!\cdots\!01}{27\!\cdots\!89}a^{17}+\frac{64\!\cdots\!50}{27\!\cdots\!89}a^{14}-\frac{14\!\cdots\!56}{27\!\cdots\!89}a^{13}+\frac{63\!\cdots\!82}{27\!\cdots\!89}a^{10}-\frac{14\!\cdots\!75}{27\!\cdots\!89}a^{9}+\frac{19\!\cdots\!28}{27\!\cdots\!89}a^{6}-\frac{46\!\cdots\!68}{27\!\cdots\!89}a^{5}+\frac{10\!\cdots\!65}{27\!\cdots\!89}a^{2}-\frac{27\!\cdots\!98}{27\!\cdots\!89}a+1$ 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:  \( 28122649019657.055 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 28122649019657.055 \cdot 3249}{8\cdot\sqrt{392893567271872510941083606170645734076396278452324169894854656}}\cr\approx \mathstrut & 0.134219656308053 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 + 49*x^32 + 932*x^28 + 8695*x^24 + 41461*x^20 + 96055*x^16 + 93536*x^12 + 28314*x^8 + 1365*x^4 + 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 + 49*x^32 + 932*x^28 + 8695*x^24 + 41461*x^20 + 96055*x^16 + 93536*x^12 + 28314*x^8 + 1365*x^4 + 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 + 49*x^32 + 932*x^28 + 8695*x^24 + 41461*x^20 + 96055*x^16 + 93536*x^12 + 28314*x^8 + 1365*x^4 + 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 + 49*x^32 + 932*x^28 + 8695*x^24 + 41461*x^20 + 96055*x^16 + 93536*x^12 + 28314*x^8 + 1365*x^4 + 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

$C_2\times C_{18}$ (as 36T2):

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 $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 3.3.361.1, \(\Q(\zeta_{8})\), 6.0.8340544.1, 6.6.66724352.1, 6.0.66724352.1, \(\Q(\zeta_{19})^+\), 12.0.284936905588473856.1, 18.0.75613185918270483380568064.1, 18.18.38713951190154487490850848768.1, 18.0.38713951190154487490850848768.1

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 R $18^{2}$ $18^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{6}$ ${\href{/padicField/11.6.0.1}{6} }^{6}$ $18^{2}$ ${\href{/padicField/17.9.0.1}{9} }^{4}$ R $18^{2}$ $18^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{18}$ ${\href{/padicField/41.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$

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 $36$$4$$9$$72$
\(19\) Copy content Toggle raw display 19.18.16.1$x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$$9$$2$$16$$C_{18}$$[\ ]_{9}^{2}$
19.18.16.1$x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$$9$$2$$16$$C_{18}$$[\ ]_{9}^{2}$