Properties

Label 18.18.925...000.1
Degree $18$
Signature $[18, 0]$
Discriminant $9.251\times 10^{32}$
Root discriminant \(67.84\)
Ramified primes $2,5,41$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_3^2 : C_4$ (as 18T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 63*x^16 + 99*x^15 + 1595*x^14 - 1797*x^13 - 20806*x^12 + 14111*x^11 + 148967*x^10 - 34860*x^9 - 572263*x^8 - 108479*x^7 + 1034639*x^6 + 563303*x^5 - 562290*x^4 - 383111*x^3 + 79922*x^2 + 66448*x + 5186)
 
gp: K = bnfinit(y^18 - 2*y^17 - 63*y^16 + 99*y^15 + 1595*y^14 - 1797*y^13 - 20806*y^12 + 14111*y^11 + 148967*y^10 - 34860*y^9 - 572263*y^8 - 108479*y^7 + 1034639*y^6 + 563303*y^5 - 562290*y^4 - 383111*y^3 + 79922*y^2 + 66448*y + 5186, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 - 63*x^16 + 99*x^15 + 1595*x^14 - 1797*x^13 - 20806*x^12 + 14111*x^11 + 148967*x^10 - 34860*x^9 - 572263*x^8 - 108479*x^7 + 1034639*x^6 + 563303*x^5 - 562290*x^4 - 383111*x^3 + 79922*x^2 + 66448*x + 5186);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 - 63*x^16 + 99*x^15 + 1595*x^14 - 1797*x^13 - 20806*x^12 + 14111*x^11 + 148967*x^10 - 34860*x^9 - 572263*x^8 - 108479*x^7 + 1034639*x^6 + 563303*x^5 - 562290*x^4 - 383111*x^3 + 79922*x^2 + 66448*x + 5186)
 

\( x^{18} - 2 x^{17} - 63 x^{16} + 99 x^{15} + 1595 x^{14} - 1797 x^{13} - 20806 x^{12} + 14111 x^{11} + 148967 x^{10} - 34860 x^{9} - 572263 x^{8} - 108479 x^{7} + 1034639 x^{6} + \cdots + 5186 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[18, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(925103102315013629321000000000000\) \(\medspace = 2^{12}\cdot 5^{12}\cdot 41^{13}\) 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:  \(67.84\)
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:  $2^{2/3}5^{2/3}41^{3/4}\approx 75.20635825846645$
Ramified primes:   \(2\), \(5\), \(41\) 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:  \(\Q(\sqrt{41}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{10}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{7}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{10}a^{12}-\frac{1}{10}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{10}a^{6}-\frac{2}{5}a^{4}+\frac{1}{10}a^{3}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{20}a^{13}-\frac{1}{20}a^{12}-\frac{1}{10}a^{11}+\frac{1}{20}a^{10}+\frac{1}{20}a^{9}-\frac{1}{5}a^{8}+\frac{7}{20}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{7}{20}a^{4}+\frac{7}{20}a^{3}+\frac{1}{5}a^{2}-\frac{1}{10}a+\frac{3}{10}$, $\frac{1}{2460}a^{14}+\frac{23}{2460}a^{13}+\frac{22}{615}a^{12}+\frac{27}{820}a^{11}+\frac{221}{2460}a^{10}-\frac{37}{1230}a^{9}-\frac{39}{164}a^{8}-\frac{49}{164}a^{7}+\frac{298}{615}a^{6}+\frac{9}{164}a^{5}-\frac{241}{492}a^{4}-\frac{83}{410}a^{3}-\frac{27}{82}a^{2}+\frac{63}{410}a+\frac{62}{615}$, $\frac{1}{4920}a^{15}-\frac{1}{4920}a^{14}-\frac{19}{984}a^{13}+\frac{1}{82}a^{12}-\frac{1}{4920}a^{11}-\frac{89}{4920}a^{10}+\frac{7}{410}a^{9}-\frac{97}{328}a^{8}+\frac{2227}{4920}a^{7}-\frac{169}{410}a^{6}-\frac{263}{4920}a^{5}+\frac{331}{1640}a^{4}-\frac{69}{328}a^{3}-\frac{223}{820}a^{2}-\frac{599}{2460}a+\frac{33}{820}$, $\frac{1}{4920}a^{16}-\frac{1}{120}a^{13}-\frac{103}{4920}a^{12}+\frac{51}{820}a^{11}+\frac{301}{4920}a^{10}+\frac{9}{328}a^{9}+\frac{421}{1230}a^{8}-\frac{251}{4920}a^{7}-\frac{281}{4920}a^{6}-\frac{289}{2460}a^{5}+\frac{27}{82}a^{4}-\frac{573}{1640}a^{3}+\frac{49}{615}a^{2}+\frac{89}{1230}a-\frac{17}{820}$, $\frac{1}{28\!\cdots\!40}a^{17}+\frac{65\!\cdots\!75}{19\!\cdots\!16}a^{16}+\frac{83\!\cdots\!31}{24\!\cdots\!20}a^{15}+\frac{36\!\cdots\!23}{96\!\cdots\!80}a^{14}+\frac{35\!\cdots\!73}{24\!\cdots\!20}a^{13}-\frac{69\!\cdots\!19}{28\!\cdots\!40}a^{12}+\frac{10\!\cdots\!53}{28\!\cdots\!40}a^{11}-\frac{34\!\cdots\!43}{72\!\cdots\!60}a^{10}+\frac{65\!\cdots\!59}{96\!\cdots\!80}a^{9}+\frac{10\!\cdots\!23}{28\!\cdots\!40}a^{8}-\frac{26\!\cdots\!52}{18\!\cdots\!65}a^{7}+\frac{48\!\cdots\!65}{19\!\cdots\!16}a^{6}-\frac{55\!\cdots\!83}{12\!\cdots\!10}a^{5}-\frac{18\!\cdots\!55}{14\!\cdots\!28}a^{4}-\frac{23\!\cdots\!97}{57\!\cdots\!48}a^{3}-\frac{12\!\cdots\!58}{18\!\cdots\!65}a^{2}+\frac{21\!\cdots\!37}{48\!\cdots\!40}a+\frac{21\!\cdots\!95}{28\!\cdots\!24}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{3}$, which has order $3$ (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:   \( -1 \)  (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:   $\frac{20\!\cdots\!53}{11\!\cdots\!40}a^{17}-\frac{37\!\cdots\!11}{11\!\cdots\!40}a^{16}-\frac{33\!\cdots\!39}{28\!\cdots\!60}a^{15}+\frac{18\!\cdots\!69}{11\!\cdots\!40}a^{14}+\frac{13\!\cdots\!00}{48\!\cdots\!01}a^{13}-\frac{10\!\cdots\!73}{38\!\cdots\!80}a^{12}-\frac{28\!\cdots\!53}{76\!\cdots\!16}a^{11}+\frac{29\!\cdots\!43}{14\!\cdots\!30}a^{10}+\frac{99\!\cdots\!39}{38\!\cdots\!80}a^{9}-\frac{10\!\cdots\!07}{38\!\cdots\!80}a^{8}-\frac{25\!\cdots\!17}{28\!\cdots\!60}a^{7}-\frac{38\!\cdots\!81}{11\!\cdots\!40}a^{6}+\frac{23\!\cdots\!59}{24\!\cdots\!05}a^{5}+\frac{26\!\cdots\!69}{23\!\cdots\!48}a^{4}+\frac{95\!\cdots\!67}{11\!\cdots\!40}a^{3}+\frac{10\!\cdots\!89}{48\!\cdots\!01}a^{2}-\frac{74\!\cdots\!67}{19\!\cdots\!40}a-\frac{33\!\cdots\!57}{57\!\cdots\!20}$, $\frac{44\!\cdots\!31}{14\!\cdots\!20}a^{17}-\frac{41\!\cdots\!31}{48\!\cdots\!40}a^{16}-\frac{68\!\cdots\!49}{36\!\cdots\!30}a^{15}+\frac{13\!\cdots\!85}{28\!\cdots\!24}a^{14}+\frac{33\!\cdots\!13}{72\!\cdots\!60}a^{13}-\frac{13\!\cdots\!49}{14\!\cdots\!20}a^{12}-\frac{27\!\cdots\!53}{48\!\cdots\!40}a^{11}+\frac{12\!\cdots\!79}{14\!\cdots\!12}a^{10}+\frac{45\!\cdots\!37}{11\!\cdots\!40}a^{9}-\frac{56\!\cdots\!97}{14\!\cdots\!20}a^{8}-\frac{10\!\cdots\!23}{72\!\cdots\!60}a^{7}+\frac{32\!\cdots\!83}{48\!\cdots\!40}a^{6}+\frac{42\!\cdots\!01}{17\!\cdots\!60}a^{5}-\frac{12\!\cdots\!31}{14\!\cdots\!20}a^{4}-\frac{18\!\cdots\!39}{14\!\cdots\!20}a^{3}-\frac{56\!\cdots\!19}{36\!\cdots\!30}a^{2}+\frac{12\!\cdots\!79}{72\!\cdots\!60}a+\frac{24\!\cdots\!57}{72\!\cdots\!60}$, $\frac{40\!\cdots\!07}{72\!\cdots\!06}a^{17}-\frac{40\!\cdots\!75}{28\!\cdots\!24}a^{16}-\frac{83\!\cdots\!33}{24\!\cdots\!20}a^{15}+\frac{12\!\cdots\!88}{18\!\cdots\!65}a^{14}+\frac{12\!\cdots\!73}{14\!\cdots\!20}a^{13}-\frac{20\!\cdots\!61}{14\!\cdots\!20}a^{12}-\frac{19\!\cdots\!93}{17\!\cdots\!60}a^{11}+\frac{18\!\cdots\!79}{14\!\cdots\!20}a^{10}+\frac{11\!\cdots\!41}{14\!\cdots\!20}a^{9}-\frac{64\!\cdots\!29}{12\!\cdots\!10}a^{8}-\frac{13\!\cdots\!59}{48\!\cdots\!40}a^{7}+\frac{63\!\cdots\!79}{96\!\cdots\!08}a^{6}+\frac{38\!\cdots\!59}{72\!\cdots\!60}a^{5}+\frac{41\!\cdots\!71}{48\!\cdots\!04}a^{4}-\frac{99\!\cdots\!27}{28\!\cdots\!24}a^{3}-\frac{46\!\cdots\!78}{60\!\cdots\!55}a^{2}+\frac{16\!\cdots\!42}{18\!\cdots\!65}a+\frac{55\!\cdots\!71}{24\!\cdots\!20}$, $\frac{25\!\cdots\!21}{14\!\cdots\!20}a^{17}-\frac{34\!\cdots\!21}{96\!\cdots\!08}a^{16}-\frac{53\!\cdots\!47}{48\!\cdots\!40}a^{15}+\frac{12\!\cdots\!07}{72\!\cdots\!60}a^{14}+\frac{40\!\cdots\!81}{14\!\cdots\!20}a^{13}-\frac{85\!\cdots\!81}{28\!\cdots\!24}a^{12}-\frac{25\!\cdots\!57}{72\!\cdots\!60}a^{11}+\frac{29\!\cdots\!23}{14\!\cdots\!20}a^{10}+\frac{35\!\cdots\!43}{14\!\cdots\!20}a^{9}-\frac{14\!\cdots\!51}{72\!\cdots\!60}a^{8}-\frac{12\!\cdots\!21}{14\!\cdots\!20}a^{7}-\frac{51\!\cdots\!99}{14\!\cdots\!20}a^{6}+\frac{11\!\cdots\!33}{96\!\cdots\!08}a^{5}+\frac{83\!\cdots\!63}{72\!\cdots\!60}a^{4}-\frac{53\!\cdots\!83}{18\!\cdots\!65}a^{3}-\frac{24\!\cdots\!79}{72\!\cdots\!60}a^{2}-\frac{56\!\cdots\!07}{60\!\cdots\!55}a-\frac{73\!\cdots\!29}{36\!\cdots\!30}$, $\frac{12\!\cdots\!07}{36\!\cdots\!30}a^{17}-\frac{15\!\cdots\!59}{14\!\cdots\!20}a^{16}-\frac{30\!\cdots\!11}{14\!\cdots\!20}a^{15}+\frac{85\!\cdots\!51}{14\!\cdots\!20}a^{14}+\frac{36\!\cdots\!19}{72\!\cdots\!60}a^{13}-\frac{11\!\cdots\!15}{96\!\cdots\!08}a^{12}-\frac{90\!\cdots\!07}{14\!\cdots\!20}a^{11}+\frac{87\!\cdots\!01}{72\!\cdots\!60}a^{10}+\frac{20\!\cdots\!19}{48\!\cdots\!40}a^{9}-\frac{29\!\cdots\!69}{48\!\cdots\!40}a^{8}-\frac{11\!\cdots\!03}{72\!\cdots\!60}a^{7}+\frac{19\!\cdots\!89}{14\!\cdots\!20}a^{6}+\frac{41\!\cdots\!79}{14\!\cdots\!20}a^{5}-\frac{14\!\cdots\!09}{14\!\cdots\!20}a^{4}-\frac{14\!\cdots\!11}{72\!\cdots\!06}a^{3}+\frac{45\!\cdots\!61}{24\!\cdots\!20}a^{2}+\frac{33\!\cdots\!21}{72\!\cdots\!60}a+\frac{57\!\cdots\!23}{36\!\cdots\!30}$, $\frac{96\!\cdots\!27}{28\!\cdots\!40}a^{17}-\frac{57\!\cdots\!21}{57\!\cdots\!48}a^{16}-\frac{48\!\cdots\!05}{24\!\cdots\!02}a^{15}+\frac{30\!\cdots\!91}{57\!\cdots\!48}a^{14}+\frac{35\!\cdots\!03}{72\!\cdots\!60}a^{13}-\frac{30\!\cdots\!77}{28\!\cdots\!40}a^{12}-\frac{17\!\cdots\!57}{28\!\cdots\!40}a^{11}+\frac{14\!\cdots\!03}{14\!\cdots\!12}a^{10}+\frac{11\!\cdots\!83}{28\!\cdots\!40}a^{9}-\frac{92\!\cdots\!53}{19\!\cdots\!16}a^{8}-\frac{17\!\cdots\!19}{12\!\cdots\!10}a^{7}+\frac{90\!\cdots\!71}{96\!\cdots\!80}a^{6}+\frac{18\!\cdots\!13}{72\!\cdots\!60}a^{5}-\frac{62\!\cdots\!07}{19\!\cdots\!16}a^{4}-\frac{86\!\cdots\!55}{57\!\cdots\!48}a^{3}-\frac{94\!\cdots\!37}{12\!\cdots\!10}a^{2}+\frac{38\!\cdots\!17}{14\!\cdots\!20}a+\frac{20\!\cdots\!87}{48\!\cdots\!40}$, $\frac{22\!\cdots\!83}{11\!\cdots\!80}a^{17}-\frac{13\!\cdots\!03}{33\!\cdots\!40}a^{16}-\frac{71\!\cdots\!93}{56\!\cdots\!40}a^{15}+\frac{66\!\cdots\!43}{33\!\cdots\!40}a^{14}+\frac{26\!\cdots\!79}{84\!\cdots\!10}a^{13}-\frac{11\!\cdots\!73}{33\!\cdots\!40}a^{12}-\frac{91\!\cdots\!47}{22\!\cdots\!56}a^{11}+\frac{96\!\cdots\!56}{42\!\cdots\!55}a^{10}+\frac{19\!\cdots\!67}{67\!\cdots\!68}a^{9}-\frac{44\!\cdots\!03}{33\!\cdots\!40}a^{8}-\frac{42\!\cdots\!34}{42\!\cdots\!55}a^{7}-\frac{54\!\cdots\!17}{11\!\cdots\!80}a^{6}+\frac{23\!\cdots\!39}{16\!\cdots\!20}a^{5}+\frac{51\!\cdots\!77}{33\!\cdots\!40}a^{4}+\frac{20\!\cdots\!29}{11\!\cdots\!80}a^{3}-\frac{39\!\cdots\!41}{84\!\cdots\!10}a^{2}-\frac{51\!\cdots\!33}{16\!\cdots\!20}a-\frac{25\!\cdots\!49}{41\!\cdots\!20}$, $\frac{57\!\cdots\!39}{14\!\cdots\!20}a^{17}-\frac{81\!\cdots\!99}{72\!\cdots\!60}a^{16}-\frac{17\!\cdots\!19}{72\!\cdots\!60}a^{15}+\frac{56\!\cdots\!81}{96\!\cdots\!08}a^{14}+\frac{83\!\cdots\!79}{14\!\cdots\!20}a^{13}-\frac{71\!\cdots\!92}{60\!\cdots\!55}a^{12}-\frac{10\!\cdots\!63}{14\!\cdots\!20}a^{11}+\frac{16\!\cdots\!61}{14\!\cdots\!20}a^{10}+\frac{34\!\cdots\!37}{72\!\cdots\!60}a^{9}-\frac{76\!\cdots\!31}{14\!\cdots\!20}a^{8}-\frac{25\!\cdots\!03}{14\!\cdots\!20}a^{7}+\frac{59\!\cdots\!79}{60\!\cdots\!55}a^{6}+\frac{14\!\cdots\!17}{48\!\cdots\!04}a^{5}-\frac{41\!\cdots\!11}{14\!\cdots\!20}a^{4}-\frac{58\!\cdots\!71}{36\!\cdots\!30}a^{3}-\frac{81\!\cdots\!51}{18\!\cdots\!65}a^{2}+\frac{57\!\cdots\!91}{24\!\cdots\!20}a+\frac{53\!\cdots\!73}{18\!\cdots\!65}$, $\frac{14\!\cdots\!27}{18\!\cdots\!65}a^{17}+\frac{10\!\cdots\!29}{48\!\cdots\!40}a^{16}-\frac{10\!\cdots\!57}{28\!\cdots\!24}a^{15}-\frac{51\!\cdots\!97}{14\!\cdots\!20}a^{14}+\frac{50\!\cdots\!77}{72\!\cdots\!60}a^{13}+\frac{22\!\cdots\!63}{14\!\cdots\!20}a^{12}-\frac{71\!\cdots\!05}{96\!\cdots\!08}a^{11}-\frac{22\!\cdots\!43}{72\!\cdots\!60}a^{10}+\frac{28\!\cdots\!67}{48\!\cdots\!40}a^{9}+\frac{45\!\cdots\!41}{14\!\cdots\!20}a^{8}-\frac{27\!\cdots\!37}{72\!\cdots\!60}a^{7}-\frac{76\!\cdots\!97}{48\!\cdots\!40}a^{6}+\frac{21\!\cdots\!83}{14\!\cdots\!20}a^{5}+\frac{51\!\cdots\!89}{14\!\cdots\!20}a^{4}-\frac{19\!\cdots\!69}{72\!\cdots\!60}a^{3}-\frac{16\!\cdots\!17}{72\!\cdots\!60}a^{2}+\frac{17\!\cdots\!27}{72\!\cdots\!60}a+\frac{61\!\cdots\!49}{18\!\cdots\!65}$, $\frac{66\!\cdots\!11}{18\!\cdots\!65}a^{17}-\frac{18\!\cdots\!33}{12\!\cdots\!51}a^{16}-\frac{99\!\cdots\!53}{48\!\cdots\!04}a^{15}+\frac{30\!\cdots\!47}{36\!\cdots\!30}a^{14}+\frac{65\!\cdots\!21}{14\!\cdots\!12}a^{13}-\frac{12\!\cdots\!11}{72\!\cdots\!60}a^{12}-\frac{89\!\cdots\!77}{18\!\cdots\!65}a^{11}+\frac{26\!\cdots\!17}{14\!\cdots\!12}a^{10}+\frac{20\!\cdots\!49}{72\!\cdots\!60}a^{9}-\frac{33\!\cdots\!51}{36\!\cdots\!30}a^{8}-\frac{61\!\cdots\!33}{72\!\cdots\!60}a^{7}+\frac{31\!\cdots\!51}{14\!\cdots\!12}a^{6}+\frac{17\!\cdots\!62}{14\!\cdots\!55}a^{5}-\frac{12\!\cdots\!73}{72\!\cdots\!60}a^{4}-\frac{19\!\cdots\!99}{36\!\cdots\!53}a^{3}+\frac{80\!\cdots\!34}{18\!\cdots\!65}a^{2}+\frac{59\!\cdots\!87}{12\!\cdots\!10}a+\frac{28\!\cdots\!22}{18\!\cdots\!65}$, $\frac{21\!\cdots\!31}{57\!\cdots\!48}a^{17}-\frac{30\!\cdots\!11}{28\!\cdots\!40}a^{16}-\frac{32\!\cdots\!79}{14\!\cdots\!20}a^{15}+\frac{16\!\cdots\!49}{28\!\cdots\!40}a^{14}+\frac{66\!\cdots\!53}{12\!\cdots\!10}a^{13}-\frac{11\!\cdots\!09}{96\!\cdots\!80}a^{12}-\frac{66\!\cdots\!97}{96\!\cdots\!80}a^{11}+\frac{40\!\cdots\!41}{36\!\cdots\!53}a^{10}+\frac{13\!\cdots\!37}{28\!\cdots\!40}a^{9}-\frac{15\!\cdots\!21}{28\!\cdots\!40}a^{8}-\frac{85\!\cdots\!59}{48\!\cdots\!04}a^{7}+\frac{10\!\cdots\!29}{96\!\cdots\!80}a^{6}+\frac{49\!\cdots\!97}{14\!\cdots\!20}a^{5}-\frac{15\!\cdots\!07}{28\!\cdots\!40}a^{4}-\frac{74\!\cdots\!87}{28\!\cdots\!40}a^{3}-\frac{89\!\cdots\!11}{72\!\cdots\!60}a^{2}+\frac{92\!\cdots\!07}{14\!\cdots\!20}a+\frac{11\!\cdots\!89}{14\!\cdots\!20}$, $\frac{43\!\cdots\!89}{60\!\cdots\!55}a^{17}-\frac{54\!\cdots\!87}{24\!\cdots\!20}a^{16}-\frac{15\!\cdots\!00}{36\!\cdots\!53}a^{15}+\frac{22\!\cdots\!04}{18\!\cdots\!65}a^{14}+\frac{73\!\cdots\!11}{72\!\cdots\!06}a^{13}-\frac{15\!\cdots\!32}{60\!\cdots\!55}a^{12}-\frac{44\!\cdots\!83}{36\!\cdots\!53}a^{11}+\frac{45\!\cdots\!22}{18\!\cdots\!65}a^{10}+\frac{48\!\cdots\!76}{60\!\cdots\!55}a^{9}-\frac{15\!\cdots\!98}{12\!\cdots\!51}a^{8}-\frac{10\!\cdots\!63}{36\!\cdots\!30}a^{7}+\frac{17\!\cdots\!48}{60\!\cdots\!55}a^{6}+\frac{86\!\cdots\!59}{18\!\cdots\!65}a^{5}-\frac{57\!\cdots\!59}{24\!\cdots\!20}a^{4}-\frac{31\!\cdots\!80}{12\!\cdots\!51}a^{3}+\frac{54\!\cdots\!68}{60\!\cdots\!55}a^{2}+\frac{79\!\cdots\!09}{36\!\cdots\!30}a-\frac{14\!\cdots\!41}{60\!\cdots\!55}$, $\frac{30\!\cdots\!99}{70\!\cdots\!40}a^{17}-\frac{11\!\cdots\!43}{96\!\cdots\!80}a^{16}-\frac{18\!\cdots\!47}{72\!\cdots\!60}a^{15}+\frac{17\!\cdots\!99}{28\!\cdots\!40}a^{14}+\frac{11\!\cdots\!63}{18\!\cdots\!65}a^{13}-\frac{34\!\cdots\!41}{28\!\cdots\!40}a^{12}-\frac{15\!\cdots\!03}{19\!\cdots\!16}a^{11}+\frac{40\!\cdots\!81}{36\!\cdots\!30}a^{10}+\frac{52\!\cdots\!73}{96\!\cdots\!80}a^{9}-\frac{14\!\cdots\!67}{28\!\cdots\!40}a^{8}-\frac{14\!\cdots\!63}{72\!\cdots\!60}a^{7}+\frac{80\!\cdots\!99}{96\!\cdots\!80}a^{6}+\frac{60\!\cdots\!34}{18\!\cdots\!65}a^{5}+\frac{13\!\cdots\!87}{28\!\cdots\!40}a^{4}-\frac{98\!\cdots\!23}{57\!\cdots\!48}a^{3}-\frac{16\!\cdots\!59}{18\!\cdots\!65}a^{2}+\frac{72\!\cdots\!37}{28\!\cdots\!24}a+\frac{66\!\cdots\!49}{28\!\cdots\!24}$, $\frac{16\!\cdots\!99}{28\!\cdots\!40}a^{17}-\frac{45\!\cdots\!99}{28\!\cdots\!40}a^{16}-\frac{48\!\cdots\!69}{14\!\cdots\!20}a^{15}+\frac{23\!\cdots\!93}{28\!\cdots\!40}a^{14}+\frac{29\!\cdots\!11}{36\!\cdots\!30}a^{13}-\frac{47\!\cdots\!67}{28\!\cdots\!40}a^{12}-\frac{96\!\cdots\!21}{96\!\cdots\!80}a^{11}+\frac{28\!\cdots\!72}{18\!\cdots\!65}a^{10}+\frac{19\!\cdots\!33}{28\!\cdots\!40}a^{9}-\frac{66\!\cdots\!19}{96\!\cdots\!80}a^{8}-\frac{17\!\cdots\!69}{72\!\cdots\!60}a^{7}+\frac{32\!\cdots\!91}{28\!\cdots\!40}a^{6}+\frac{20\!\cdots\!29}{48\!\cdots\!40}a^{5}+\frac{42\!\cdots\!73}{28\!\cdots\!40}a^{4}-\frac{13\!\cdots\!79}{57\!\cdots\!48}a^{3}-\frac{82\!\cdots\!59}{24\!\cdots\!20}a^{2}+\frac{19\!\cdots\!01}{48\!\cdots\!40}a+\frac{77\!\cdots\!57}{14\!\cdots\!20}$, $\frac{11\!\cdots\!77}{96\!\cdots\!80}a^{17}-\frac{11\!\cdots\!87}{28\!\cdots\!40}a^{16}-\frac{52\!\cdots\!79}{72\!\cdots\!60}a^{15}+\frac{21\!\cdots\!33}{96\!\cdots\!80}a^{14}+\frac{24\!\cdots\!59}{14\!\cdots\!20}a^{13}-\frac{88\!\cdots\!57}{19\!\cdots\!16}a^{12}-\frac{60\!\cdots\!09}{28\!\cdots\!40}a^{11}+\frac{66\!\cdots\!29}{14\!\cdots\!20}a^{10}+\frac{40\!\cdots\!21}{28\!\cdots\!40}a^{9}-\frac{65\!\cdots\!51}{28\!\cdots\!40}a^{8}-\frac{14\!\cdots\!25}{28\!\cdots\!24}a^{7}+\frac{13\!\cdots\!83}{28\!\cdots\!40}a^{6}+\frac{62\!\cdots\!01}{72\!\cdots\!60}a^{5}-\frac{18\!\cdots\!61}{70\!\cdots\!40}a^{4}-\frac{42\!\cdots\!99}{96\!\cdots\!80}a^{3}+\frac{67\!\cdots\!67}{88\!\cdots\!33}a^{2}+\frac{25\!\cdots\!81}{28\!\cdots\!24}a+\frac{12\!\cdots\!71}{14\!\cdots\!20}$, $\frac{11\!\cdots\!47}{28\!\cdots\!40}a^{17}-\frac{19\!\cdots\!93}{28\!\cdots\!40}a^{16}-\frac{31\!\cdots\!19}{12\!\cdots\!10}a^{15}+\frac{44\!\cdots\!89}{96\!\cdots\!80}a^{14}+\frac{14\!\cdots\!23}{18\!\cdots\!65}a^{13}-\frac{35\!\cdots\!13}{28\!\cdots\!40}a^{12}-\frac{46\!\cdots\!49}{28\!\cdots\!40}a^{11}+\frac{30\!\cdots\!68}{18\!\cdots\!65}a^{10}+\frac{19\!\cdots\!41}{96\!\cdots\!80}a^{9}-\frac{37\!\cdots\!79}{28\!\cdots\!40}a^{8}-\frac{21\!\cdots\!71}{14\!\cdots\!12}a^{7}+\frac{14\!\cdots\!01}{28\!\cdots\!40}a^{6}+\frac{21\!\cdots\!31}{36\!\cdots\!30}a^{5}-\frac{23\!\cdots\!57}{28\!\cdots\!40}a^{4}-\frac{61\!\cdots\!51}{57\!\cdots\!48}a^{3}+\frac{36\!\cdots\!66}{18\!\cdots\!65}a^{2}+\frac{57\!\cdots\!61}{14\!\cdots\!20}a+\frac{50\!\cdots\!81}{14\!\cdots\!20}$, $\frac{25\!\cdots\!81}{70\!\cdots\!40}a^{17}-\frac{77\!\cdots\!83}{28\!\cdots\!40}a^{16}-\frac{10\!\cdots\!62}{60\!\cdots\!55}a^{15}+\frac{15\!\cdots\!63}{96\!\cdots\!80}a^{14}+\frac{12\!\cdots\!91}{36\!\cdots\!30}a^{13}-\frac{71\!\cdots\!73}{19\!\cdots\!16}a^{12}-\frac{83\!\cdots\!31}{28\!\cdots\!40}a^{11}+\frac{53\!\cdots\!21}{12\!\cdots\!10}a^{10}+\frac{12\!\cdots\!59}{96\!\cdots\!80}a^{9}-\frac{55\!\cdots\!55}{19\!\cdots\!16}a^{8}-\frac{11\!\cdots\!41}{24\!\cdots\!20}a^{7}+\frac{28\!\cdots\!59}{28\!\cdots\!40}a^{6}+\frac{96\!\cdots\!91}{36\!\cdots\!30}a^{5}-\frac{43\!\cdots\!39}{28\!\cdots\!40}a^{4}-\frac{20\!\cdots\!77}{28\!\cdots\!40}a^{3}+\frac{32\!\cdots\!72}{60\!\cdots\!55}a^{2}+\frac{39\!\cdots\!47}{14\!\cdots\!20}a-\frac{21\!\cdots\!89}{14\!\cdots\!20}$ 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:  \( 156794220435 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 156794220435 \cdot 3}{2\cdot\sqrt{925103102315013629321000000000000}}\cr\approx \mathstrut & 2.02705794036859 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 63*x^16 + 99*x^15 + 1595*x^14 - 1797*x^13 - 20806*x^12 + 14111*x^11 + 148967*x^10 - 34860*x^9 - 572263*x^8 - 108479*x^7 + 1034639*x^6 + 563303*x^5 - 562290*x^4 - 383111*x^3 + 79922*x^2 + 66448*x + 5186)
 
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^18 - 2*x^17 - 63*x^16 + 99*x^15 + 1595*x^14 - 1797*x^13 - 20806*x^12 + 14111*x^11 + 148967*x^10 - 34860*x^9 - 572263*x^8 - 108479*x^7 + 1034639*x^6 + 563303*x^5 - 562290*x^4 - 383111*x^3 + 79922*x^2 + 66448*x + 5186, 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^18 - 2*x^17 - 63*x^16 + 99*x^15 + 1595*x^14 - 1797*x^13 - 20806*x^12 + 14111*x^11 + 148967*x^10 - 34860*x^9 - 572263*x^8 - 108479*x^7 + 1034639*x^6 + 563303*x^5 - 562290*x^4 - 383111*x^3 + 79922*x^2 + 66448*x + 5186);
 
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^18 - 2*x^17 - 63*x^16 + 99*x^15 + 1595*x^14 - 1797*x^13 - 20806*x^12 + 14111*x^11 + 148967*x^10 - 34860*x^9 - 572263*x^8 - 108479*x^7 + 1034639*x^6 + 563303*x^5 - 562290*x^4 - 383111*x^3 + 79922*x^2 + 66448*x + 5186);
 
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

$C_3^2:C_4$ (as 18T10):

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)
 
A solvable group of order 36
The 6 conjugacy class representatives for $C_3^2 : C_4$
Character table for $C_3^2 : C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 6.6.7064402500.1 x2, 6.6.1130304400.1 x2, 9.9.4750104241000000.1

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]
 

Sibling fields

Galois closure: data not computed
Degree 6 siblings: 6.6.7064402500.1, 6.6.1130304400.1
Degree 9 sibling: 9.9.4750104241000000.1
Degree 12 siblings: deg 12, deg 12
Minimal sibling: 6.6.1130304400.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }$ R ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ R ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(5\) Copy content Toggle raw display 5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(41\) Copy content Toggle raw display 41.2.1.1$x^{2} + 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.3.1$x^{4} + 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} + 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} + 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} + 41$$4$$1$$3$$C_4$$[\ ]_{4}$