Properties

Label 18.2.402...137.1
Degree $18$
Signature $[2, 8]$
Discriminant $4.025\times 10^{40}$
Root discriminant \(180.23\)
Ramified prime $17$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PGL(2,17)$ (as 18T468)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 17*x^16 + 17*x^15 - 935*x^14 + 799*x^13 + 9231*x^12 - 41463*x^11 + 192780*x^10 + 291686*x^9 - 390014*x^8 + 6132223*x^7 - 3955645*x^6 + 2916112*x^5 + 45030739*x^4 - 94452714*x^3 + 184016925*x^2 - 141466230*x + 113422599)
 
gp: K = bnfinit(y^18 - 7*y^17 + 17*y^16 + 17*y^15 - 935*y^14 + 799*y^13 + 9231*y^12 - 41463*y^11 + 192780*y^10 + 291686*y^9 - 390014*y^8 + 6132223*y^7 - 3955645*y^6 + 2916112*y^5 + 45030739*y^4 - 94452714*y^3 + 184016925*y^2 - 141466230*y + 113422599, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 7*x^17 + 17*x^16 + 17*x^15 - 935*x^14 + 799*x^13 + 9231*x^12 - 41463*x^11 + 192780*x^10 + 291686*x^9 - 390014*x^8 + 6132223*x^7 - 3955645*x^6 + 2916112*x^5 + 45030739*x^4 - 94452714*x^3 + 184016925*x^2 - 141466230*x + 113422599);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 7*x^17 + 17*x^16 + 17*x^15 - 935*x^14 + 799*x^13 + 9231*x^12 - 41463*x^11 + 192780*x^10 + 291686*x^9 - 390014*x^8 + 6132223*x^7 - 3955645*x^6 + 2916112*x^5 + 45030739*x^4 - 94452714*x^3 + 184016925*x^2 - 141466230*x + 113422599)
 

\( x^{18} - 7 x^{17} + 17 x^{16} + 17 x^{15} - 935 x^{14} + 799 x^{13} + 9231 x^{12} - 41463 x^{11} + 192780 x^{10} + 291686 x^{9} - 390014 x^{8} + 6132223 x^{7} + \cdots + 113422599 \) 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:  $[2, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(40254497110927943179349807054456171205137\) \(\medspace = 17^{33}\) 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:  \(180.23\)
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:  $17^{543/272}\approx 286.0053344539552$
Ramified primes:   \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{1}{9}a^{5}-\frac{2}{9}a^{4}-\frac{4}{9}a^{3}-\frac{2}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{2}{9}a^{4}+\frac{2}{9}a^{3}-\frac{1}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{27}a^{13}-\frac{1}{27}a^{11}-\frac{1}{27}a^{10}+\frac{1}{27}a^{9}+\frac{1}{27}a^{8}-\frac{1}{27}a^{7}-\frac{1}{27}a^{6}+\frac{4}{27}a^{5}-\frac{1}{3}a^{4}-\frac{1}{27}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{27}a^{14}-\frac{1}{27}a^{12}-\frac{1}{27}a^{11}+\frac{1}{27}a^{10}+\frac{1}{27}a^{9}-\frac{1}{27}a^{8}-\frac{1}{27}a^{7}+\frac{4}{27}a^{6}-\frac{10}{27}a^{4}+\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{27}a^{15}-\frac{1}{27}a^{12}+\frac{1}{9}a^{7}-\frac{1}{27}a^{6}+\frac{1}{9}a^{5}-\frac{2}{9}a^{4}-\frac{1}{27}a^{3}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{16}-\frac{1}{27}a^{11}-\frac{1}{27}a^{10}+\frac{1}{27}a^{9}+\frac{4}{27}a^{8}-\frac{2}{27}a^{7}+\frac{2}{27}a^{6}-\frac{2}{27}a^{5}-\frac{10}{27}a^{4}-\frac{7}{27}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{70\!\cdots\!37}a^{17}+\frac{33\!\cdots\!81}{70\!\cdots\!37}a^{16}+\frac{11\!\cdots\!11}{70\!\cdots\!37}a^{15}+\frac{63\!\cdots\!48}{70\!\cdots\!37}a^{14}-\frac{74\!\cdots\!10}{70\!\cdots\!37}a^{13}+\frac{26\!\cdots\!00}{70\!\cdots\!37}a^{12}+\frac{10\!\cdots\!69}{87\!\cdots\!77}a^{11}-\frac{12\!\cdots\!35}{78\!\cdots\!93}a^{10}+\frac{53\!\cdots\!04}{78\!\cdots\!93}a^{9}+\frac{48\!\cdots\!93}{70\!\cdots\!37}a^{8}+\frac{57\!\cdots\!46}{70\!\cdots\!37}a^{7}+\frac{18\!\cdots\!23}{70\!\cdots\!37}a^{6}+\frac{60\!\cdots\!55}{70\!\cdots\!37}a^{5}-\frac{34\!\cdots\!51}{70\!\cdots\!37}a^{4}-\frac{50\!\cdots\!74}{70\!\cdots\!37}a^{3}-\frac{39\!\cdots\!75}{23\!\cdots\!79}a^{2}+\frac{53\!\cdots\!13}{26\!\cdots\!31}a+\frac{18\!\cdots\!57}{26\!\cdots\!31}$ 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:  $3$

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:   $\frac{28\!\cdots\!65}{70\!\cdots\!37}a^{17}-\frac{30\!\cdots\!07}{70\!\cdots\!37}a^{16}+\frac{16\!\cdots\!38}{70\!\cdots\!37}a^{15}+\frac{21\!\cdots\!13}{70\!\cdots\!37}a^{14}-\frac{26\!\cdots\!58}{70\!\cdots\!37}a^{13}+\frac{17\!\cdots\!96}{70\!\cdots\!37}a^{12}+\frac{20\!\cdots\!86}{78\!\cdots\!93}a^{11}-\frac{44\!\cdots\!28}{87\!\cdots\!77}a^{10}+\frac{80\!\cdots\!56}{78\!\cdots\!93}a^{9}-\frac{25\!\cdots\!88}{70\!\cdots\!37}a^{8}-\frac{13\!\cdots\!23}{70\!\cdots\!37}a^{7}+\frac{12\!\cdots\!13}{70\!\cdots\!37}a^{6}-\frac{10\!\cdots\!12}{70\!\cdots\!37}a^{5}-\frac{56\!\cdots\!68}{70\!\cdots\!37}a^{4}+\frac{24\!\cdots\!19}{70\!\cdots\!37}a^{3}-\frac{31\!\cdots\!89}{23\!\cdots\!79}a^{2}+\frac{43\!\cdots\!28}{26\!\cdots\!31}a-\frac{40\!\cdots\!57}{26\!\cdots\!31}$, $\frac{14\!\cdots\!08}{70\!\cdots\!37}a^{17}-\frac{94\!\cdots\!00}{70\!\cdots\!37}a^{16}-\frac{26\!\cdots\!52}{70\!\cdots\!37}a^{15}+\frac{61\!\cdots\!63}{70\!\cdots\!37}a^{14}-\frac{51\!\cdots\!15}{70\!\cdots\!37}a^{13}+\frac{18\!\cdots\!11}{70\!\cdots\!37}a^{12}-\frac{23\!\cdots\!48}{87\!\cdots\!77}a^{11}-\frac{26\!\cdots\!90}{26\!\cdots\!31}a^{10}+\frac{64\!\cdots\!56}{78\!\cdots\!93}a^{9}-\frac{16\!\cdots\!70}{70\!\cdots\!37}a^{8}+\frac{25\!\cdots\!70}{70\!\cdots\!37}a^{7}+\frac{18\!\cdots\!36}{70\!\cdots\!37}a^{6}-\frac{16\!\cdots\!83}{70\!\cdots\!37}a^{5}+\frac{48\!\cdots\!61}{70\!\cdots\!37}a^{4}-\frac{80\!\cdots\!68}{70\!\cdots\!37}a^{3}+\frac{32\!\cdots\!68}{23\!\cdots\!79}a^{2}-\frac{26\!\cdots\!36}{26\!\cdots\!31}a+\frac{13\!\cdots\!15}{26\!\cdots\!31}$, $\frac{52\!\cdots\!62}{70\!\cdots\!37}a^{17}-\frac{23\!\cdots\!05}{70\!\cdots\!37}a^{16}+\frac{87\!\cdots\!01}{70\!\cdots\!37}a^{15}+\frac{14\!\cdots\!78}{70\!\cdots\!37}a^{14}-\frac{14\!\cdots\!89}{70\!\cdots\!37}a^{13}+\frac{14\!\cdots\!58}{70\!\cdots\!37}a^{12}+\frac{48\!\cdots\!71}{78\!\cdots\!93}a^{11}-\frac{29\!\cdots\!58}{78\!\cdots\!93}a^{10}-\frac{23\!\cdots\!53}{78\!\cdots\!93}a^{9}-\frac{15\!\cdots\!02}{70\!\cdots\!37}a^{8}-\frac{84\!\cdots\!17}{70\!\cdots\!37}a^{7}+\frac{61\!\cdots\!47}{70\!\cdots\!37}a^{6}-\frac{21\!\cdots\!33}{70\!\cdots\!37}a^{5}-\frac{39\!\cdots\!30}{70\!\cdots\!37}a^{4}+\frac{10\!\cdots\!26}{70\!\cdots\!37}a^{3}-\frac{93\!\cdots\!80}{23\!\cdots\!79}a^{2}+\frac{82\!\cdots\!66}{26\!\cdots\!31}a-\frac{78\!\cdots\!76}{26\!\cdots\!31}$, $\frac{14\!\cdots\!68}{23\!\cdots\!79}a^{17}-\frac{48\!\cdots\!94}{23\!\cdots\!79}a^{16}-\frac{76\!\cdots\!26}{23\!\cdots\!79}a^{15}+\frac{64\!\cdots\!81}{23\!\cdots\!79}a^{14}-\frac{10\!\cdots\!42}{23\!\cdots\!79}a^{13}-\frac{37\!\cdots\!98}{23\!\cdots\!79}a^{12}+\frac{39\!\cdots\!64}{78\!\cdots\!93}a^{11}+\frac{26\!\cdots\!80}{78\!\cdots\!93}a^{10}+\frac{39\!\cdots\!83}{78\!\cdots\!93}a^{9}+\frac{99\!\cdots\!25}{23\!\cdots\!79}a^{8}+\frac{23\!\cdots\!35}{23\!\cdots\!79}a^{7}+\frac{77\!\cdots\!79}{23\!\cdots\!79}a^{6}+\frac{13\!\cdots\!18}{23\!\cdots\!79}a^{5}+\frac{16\!\cdots\!44}{23\!\cdots\!79}a^{4}+\frac{37\!\cdots\!67}{23\!\cdots\!79}a^{3}-\frac{24\!\cdots\!29}{78\!\cdots\!93}a^{2}+\frac{56\!\cdots\!03}{29\!\cdots\!59}a-\frac{38\!\cdots\!22}{87\!\cdots\!77}$, $\frac{16\!\cdots\!84}{70\!\cdots\!37}a^{17}-\frac{10\!\cdots\!01}{70\!\cdots\!37}a^{16}+\frac{19\!\cdots\!39}{70\!\cdots\!37}a^{15}+\frac{49\!\cdots\!15}{70\!\cdots\!37}a^{14}-\frac{15\!\cdots\!48}{70\!\cdots\!37}a^{13}+\frac{43\!\cdots\!47}{70\!\cdots\!37}a^{12}+\frac{21\!\cdots\!02}{87\!\cdots\!77}a^{11}-\frac{61\!\cdots\!60}{78\!\cdots\!93}a^{10}+\frac{29\!\cdots\!97}{78\!\cdots\!93}a^{9}+\frac{65\!\cdots\!20}{70\!\cdots\!37}a^{8}-\frac{60\!\cdots\!94}{70\!\cdots\!37}a^{7}+\frac{81\!\cdots\!88}{70\!\cdots\!37}a^{6}-\frac{41\!\cdots\!46}{70\!\cdots\!37}a^{5}-\frac{12\!\cdots\!74}{70\!\cdots\!37}a^{4}+\frac{44\!\cdots\!99}{70\!\cdots\!37}a^{3}-\frac{51\!\cdots\!20}{23\!\cdots\!79}a^{2}+\frac{42\!\cdots\!67}{26\!\cdots\!31}a-\frac{51\!\cdots\!32}{26\!\cdots\!31}$, $\frac{13\!\cdots\!79}{23\!\cdots\!79}a^{17}-\frac{75\!\cdots\!09}{23\!\cdots\!79}a^{16}+\frac{63\!\cdots\!24}{23\!\cdots\!79}a^{15}+\frac{71\!\cdots\!23}{23\!\cdots\!79}a^{14}-\frac{12\!\cdots\!61}{23\!\cdots\!79}a^{13}-\frac{86\!\cdots\!10}{23\!\cdots\!79}a^{12}+\frac{21\!\cdots\!91}{29\!\cdots\!59}a^{11}-\frac{11\!\cdots\!30}{78\!\cdots\!93}a^{10}+\frac{15\!\cdots\!41}{26\!\cdots\!31}a^{9}+\frac{79\!\cdots\!60}{23\!\cdots\!79}a^{8}-\frac{25\!\cdots\!56}{23\!\cdots\!79}a^{7}+\frac{49\!\cdots\!24}{23\!\cdots\!79}a^{6}+\frac{51\!\cdots\!99}{23\!\cdots\!79}a^{5}-\frac{14\!\cdots\!82}{23\!\cdots\!79}a^{4}+\frac{38\!\cdots\!94}{23\!\cdots\!79}a^{3}-\frac{12\!\cdots\!56}{78\!\cdots\!93}a^{2}+\frac{23\!\cdots\!30}{29\!\cdots\!59}a+\frac{14\!\cdots\!06}{87\!\cdots\!77}$, $\frac{17\!\cdots\!87}{23\!\cdots\!79}a^{17}-\frac{18\!\cdots\!11}{23\!\cdots\!79}a^{16}+\frac{97\!\cdots\!02}{23\!\cdots\!79}a^{15}-\frac{31\!\cdots\!91}{23\!\cdots\!79}a^{14}-\frac{58\!\cdots\!71}{23\!\cdots\!79}a^{13}+\frac{40\!\cdots\!72}{23\!\cdots\!79}a^{12}+\frac{11\!\cdots\!58}{29\!\cdots\!59}a^{11}-\frac{83\!\cdots\!36}{26\!\cdots\!31}a^{10}+\frac{69\!\cdots\!17}{26\!\cdots\!31}a^{9}-\frac{17\!\cdots\!07}{23\!\cdots\!79}a^{8}+\frac{52\!\cdots\!75}{23\!\cdots\!79}a^{7}-\frac{57\!\cdots\!22}{23\!\cdots\!79}a^{6}+\frac{37\!\cdots\!66}{23\!\cdots\!79}a^{5}+\frac{18\!\cdots\!31}{23\!\cdots\!79}a^{4}-\frac{43\!\cdots\!10}{23\!\cdots\!79}a^{3}+\frac{26\!\cdots\!71}{78\!\cdots\!93}a^{2}-\frac{22\!\cdots\!59}{87\!\cdots\!77}a+\frac{16\!\cdots\!65}{87\!\cdots\!77}$, $\frac{17\!\cdots\!62}{70\!\cdots\!37}a^{17}-\frac{19\!\cdots\!36}{70\!\cdots\!37}a^{16}+\frac{71\!\cdots\!14}{70\!\cdots\!37}a^{15}+\frac{40\!\cdots\!65}{70\!\cdots\!37}a^{14}-\frac{22\!\cdots\!44}{70\!\cdots\!37}a^{13}+\frac{82\!\cdots\!25}{70\!\cdots\!37}a^{12}+\frac{28\!\cdots\!47}{78\!\cdots\!93}a^{11}-\frac{22\!\cdots\!54}{78\!\cdots\!93}a^{10}+\frac{17\!\cdots\!52}{26\!\cdots\!31}a^{9}+\frac{47\!\cdots\!46}{70\!\cdots\!37}a^{8}-\frac{63\!\cdots\!90}{70\!\cdots\!37}a^{7}+\frac{10\!\cdots\!84}{70\!\cdots\!37}a^{6}-\frac{84\!\cdots\!30}{70\!\cdots\!37}a^{5}-\frac{42\!\cdots\!57}{70\!\cdots\!37}a^{4}+\frac{12\!\cdots\!12}{70\!\cdots\!37}a^{3}-\frac{70\!\cdots\!28}{23\!\cdots\!79}a^{2}+\frac{65\!\cdots\!89}{26\!\cdots\!31}a-\frac{43\!\cdots\!26}{26\!\cdots\!31}$, $\frac{39\!\cdots\!01}{70\!\cdots\!37}a^{17}-\frac{14\!\cdots\!09}{70\!\cdots\!37}a^{16}+\frac{28\!\cdots\!88}{70\!\cdots\!37}a^{15}+\frac{22\!\cdots\!83}{70\!\cdots\!37}a^{14}-\frac{31\!\cdots\!64}{70\!\cdots\!37}a^{13}-\frac{67\!\cdots\!28}{70\!\cdots\!37}a^{12}+\frac{15\!\cdots\!35}{78\!\cdots\!93}a^{11}-\frac{22\!\cdots\!56}{78\!\cdots\!93}a^{10}-\frac{17\!\cdots\!47}{78\!\cdots\!93}a^{9}+\frac{45\!\cdots\!73}{70\!\cdots\!37}a^{8}-\frac{41\!\cdots\!41}{70\!\cdots\!37}a^{7}+\frac{31\!\cdots\!34}{70\!\cdots\!37}a^{6}+\frac{93\!\cdots\!50}{70\!\cdots\!37}a^{5}-\frac{30\!\cdots\!90}{70\!\cdots\!37}a^{4}+\frac{60\!\cdots\!77}{70\!\cdots\!37}a^{3}-\frac{33\!\cdots\!17}{23\!\cdots\!79}a^{2}+\frac{27\!\cdots\!78}{26\!\cdots\!31}a-\frac{19\!\cdots\!36}{26\!\cdots\!31}$ 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:  \( 45413380405400 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 45413380405400 \cdot 1}{2\cdot\sqrt{40254497110927943179349807054456171205137}}\cr\approx \mathstrut & 1.09962744387092 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 17*x^16 + 17*x^15 - 935*x^14 + 799*x^13 + 9231*x^12 - 41463*x^11 + 192780*x^10 + 291686*x^9 - 390014*x^8 + 6132223*x^7 - 3955645*x^6 + 2916112*x^5 + 45030739*x^4 - 94452714*x^3 + 184016925*x^2 - 141466230*x + 113422599)
 
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 - 7*x^17 + 17*x^16 + 17*x^15 - 935*x^14 + 799*x^13 + 9231*x^12 - 41463*x^11 + 192780*x^10 + 291686*x^9 - 390014*x^8 + 6132223*x^7 - 3955645*x^6 + 2916112*x^5 + 45030739*x^4 - 94452714*x^3 + 184016925*x^2 - 141466230*x + 113422599, 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 - 7*x^17 + 17*x^16 + 17*x^15 - 935*x^14 + 799*x^13 + 9231*x^12 - 41463*x^11 + 192780*x^10 + 291686*x^9 - 390014*x^8 + 6132223*x^7 - 3955645*x^6 + 2916112*x^5 + 45030739*x^4 - 94452714*x^3 + 184016925*x^2 - 141466230*x + 113422599);
 
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 - 7*x^17 + 17*x^16 + 17*x^15 - 935*x^14 + 799*x^13 + 9231*x^12 - 41463*x^11 + 192780*x^10 + 291686*x^9 - 390014*x^8 + 6132223*x^7 - 3955645*x^6 + 2916112*x^5 + 45030739*x^4 - 94452714*x^3 + 184016925*x^2 - 141466230*x + 113422599);
 
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

$\PGL(2,17)$ (as 18T468):

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 non-solvable group of order 4896
The 19 conjugacy class representatives for $\PGL(2,17)$
Character table for $\PGL(2,17)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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

Degree 36 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{2}$ ${\href{/padicField/3.2.0.1}{2} }^{9}$ $16{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ $18$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ R ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $16{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ $18$ $16{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ $16{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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
\(17\) Copy content Toggle raw display $\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
17.17.33.11$x^{17} + 867 x^{2} + 2890 x + 17$$17$$1$$33$$F_{17}$$[33/16]_{16}$

Additional information

This field is associated with the 17-division points on any elliptic curve in the isogeny class 17.a1.