Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 5*x^22 + 3*x^21 + 3*x^20 + 43*x^19 - 6*x^18 - 31*x^17 - 56*x^16 - 68*x^15 - 89*x^14 + 147*x^13 + 180*x^12 + 274*x^11 + 240*x^10 - 211*x^9 - 521*x^8 - 383*x^7 - 275*x^6 + 291*x^5 + 249*x^4 + 358*x^3 - 9*x^2 + 107*x + 1)
gp: K = bnfinit(y^23 - 5*y^22 + 3*y^21 + 3*y^20 + 43*y^19 - 6*y^18 - 31*y^17 - 56*y^16 - 68*y^15 - 89*y^14 + 147*y^13 + 180*y^12 + 274*y^11 + 240*y^10 - 211*y^9 - 521*y^8 - 383*y^7 - 275*y^6 + 291*y^5 + 249*y^4 + 358*y^3 - 9*y^2 + 107*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 5*x^22 + 3*x^21 + 3*x^20 + 43*x^19 - 6*x^18 - 31*x^17 - 56*x^16 - 68*x^15 - 89*x^14 + 147*x^13 + 180*x^12 + 274*x^11 + 240*x^10 - 211*x^9 - 521*x^8 - 383*x^7 - 275*x^6 + 291*x^5 + 249*x^4 + 358*x^3 - 9*x^2 + 107*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 5*x^22 + 3*x^21 + 3*x^20 + 43*x^19 - 6*x^18 - 31*x^17 - 56*x^16 - 68*x^15 - 89*x^14 + 147*x^13 + 180*x^12 + 274*x^11 + 240*x^10 - 211*x^9 - 521*x^8 - 383*x^7 - 275*x^6 + 291*x^5 + 249*x^4 + 358*x^3 - 9*x^2 + 107*x + 1)
\( x^{23} - 5 x^{22} + 3 x^{21} + 3 x^{20} + 43 x^{19} - 6 x^{18} - 31 x^{17} - 56 x^{16} - 68 x^{15} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-69781355284066743169711357773860671\)
\(\medspace = -\,1471^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.73\) | 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: | $1471^{1/2}\approx 38.353617821530214$ | ||
| Ramified primes: |
\(1471\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1471}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{189}a^{19}+\frac{5}{189}a^{18}+\frac{1}{27}a^{17}-\frac{10}{189}a^{16}-\frac{10}{63}a^{15}+\frac{1}{21}a^{14}-\frac{1}{63}a^{13}-\frac{5}{189}a^{11}-\frac{22}{189}a^{10}-\frac{8}{189}a^{9}-\frac{1}{189}a^{8}+\frac{1}{9}a^{7}-\frac{1}{3}a^{6}+\frac{4}{9}a^{5}+\frac{2}{7}a^{4}+\frac{31}{189}a^{3}-\frac{55}{189}a^{2}-\frac{89}{189}a-\frac{34}{189}$, $\frac{1}{189}a^{20}+\frac{1}{63}a^{18}-\frac{1}{63}a^{17}-\frac{1}{189}a^{16}-\frac{10}{63}a^{15}+\frac{5}{63}a^{14}+\frac{5}{63}a^{13}-\frac{5}{189}a^{12}+\frac{1}{63}a^{11}-\frac{1}{63}a^{10}+\frac{2}{21}a^{9}+\frac{5}{189}a^{8}+\frac{1}{9}a^{7}-\frac{2}{9}a^{6}+\frac{4}{63}a^{5}-\frac{50}{189}a^{4}-\frac{1}{9}a^{3}+\frac{3}{7}a^{2}+\frac{4}{63}a+\frac{23}{189}$, $\frac{1}{334341}a^{21}-\frac{28}{15921}a^{20}+\frac{187}{334341}a^{19}+\frac{566}{47763}a^{18}+\frac{5378}{111447}a^{17}+\frac{9659}{334341}a^{16}+\frac{9946}{111447}a^{15}+\frac{3455}{111447}a^{14}+\frac{40393}{334341}a^{13}-\frac{5273}{37149}a^{12}+\frac{35680}{334341}a^{11}+\frac{10859}{334341}a^{10}-\frac{5662}{111447}a^{9}+\frac{46898}{334341}a^{8}-\frac{3184}{15921}a^{7}+\frac{54226}{111447}a^{6}-\frac{101900}{334341}a^{5}-\frac{26480}{111447}a^{4}-\frac{71831}{334341}a^{3}+\frac{8819}{47763}a^{2}-\frac{9133}{111447}a+\frac{55736}{334341}$, $\frac{1}{10\!\cdots\!59}a^{22}-\frac{1239853753063}{10\!\cdots\!59}a^{21}+\frac{24\!\cdots\!30}{10\!\cdots\!59}a^{20}-\frac{12\!\cdots\!18}{10\!\cdots\!59}a^{19}+\frac{15\!\cdots\!75}{35\!\cdots\!53}a^{18}-\frac{440673393684886}{50\!\cdots\!79}a^{17}-\frac{11\!\cdots\!27}{10\!\cdots\!59}a^{16}+\frac{66\!\cdots\!66}{50\!\cdots\!79}a^{15}+\frac{59\!\cdots\!80}{10\!\cdots\!59}a^{14}-\frac{35\!\cdots\!86}{10\!\cdots\!59}a^{13}-\frac{13\!\cdots\!00}{10\!\cdots\!59}a^{12}-\frac{134903839943941}{10\!\cdots\!59}a^{11}-\frac{41\!\cdots\!45}{35\!\cdots\!53}a^{10}+\frac{22\!\cdots\!98}{38\!\cdots\!17}a^{9}-\frac{94\!\cdots\!52}{10\!\cdots\!59}a^{8}-\frac{13\!\cdots\!21}{35\!\cdots\!53}a^{7}+\frac{38\!\cdots\!37}{80\!\cdots\!43}a^{6}-\frac{38\!\cdots\!36}{10\!\cdots\!59}a^{5}-\frac{45\!\cdots\!61}{10\!\cdots\!59}a^{4}-\frac{43\!\cdots\!05}{10\!\cdots\!59}a^{3}+\frac{13\!\cdots\!23}{35\!\cdots\!53}a^{2}-\frac{81\!\cdots\!80}{35\!\cdots\!53}a-\frac{47\!\cdots\!79}{10\!\cdots\!59}$
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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{60\!\cdots\!29}{10\!\cdots\!59}a^{22}-\frac{30\!\cdots\!60}{10\!\cdots\!59}a^{21}+\frac{18\!\cdots\!52}{10\!\cdots\!59}a^{20}+\frac{29\!\cdots\!36}{10\!\cdots\!59}a^{19}+\frac{80\!\cdots\!81}{35\!\cdots\!53}a^{18}-\frac{27\!\cdots\!66}{35\!\cdots\!53}a^{17}-\frac{24\!\cdots\!55}{10\!\cdots\!59}a^{16}-\frac{69\!\cdots\!79}{35\!\cdots\!53}a^{15}-\frac{25\!\cdots\!43}{10\!\cdots\!59}a^{14}-\frac{68\!\cdots\!76}{10\!\cdots\!59}a^{13}+\frac{27\!\cdots\!64}{36\!\cdots\!71}a^{12}+\frac{11\!\cdots\!56}{10\!\cdots\!59}a^{11}+\frac{53\!\cdots\!46}{50\!\cdots\!79}a^{10}+\frac{47\!\cdots\!66}{40\!\cdots\!19}a^{9}-\frac{83\!\cdots\!19}{15\!\cdots\!37}a^{8}-\frac{93\!\cdots\!60}{35\!\cdots\!53}a^{7}-\frac{95\!\cdots\!21}{80\!\cdots\!43}a^{6}-\frac{12\!\cdots\!06}{10\!\cdots\!59}a^{5}+\frac{19\!\cdots\!50}{10\!\cdots\!59}a^{4}+\frac{31\!\cdots\!39}{10\!\cdots\!59}a^{3}+\frac{65\!\cdots\!84}{35\!\cdots\!53}a^{2}+\frac{28\!\cdots\!06}{35\!\cdots\!53}a+\frac{19\!\cdots\!41}{15\!\cdots\!37}$, $\frac{13\!\cdots\!92}{10\!\cdots\!59}a^{22}-\frac{70\!\cdots\!91}{15\!\cdots\!37}a^{21}-\frac{63\!\cdots\!70}{10\!\cdots\!59}a^{20}+\frac{15\!\cdots\!72}{10\!\cdots\!59}a^{19}+\frac{22\!\cdots\!28}{35\!\cdots\!53}a^{18}+\frac{17\!\cdots\!77}{35\!\cdots\!53}a^{17}-\frac{12\!\cdots\!96}{10\!\cdots\!59}a^{16}-\frac{77\!\cdots\!14}{50\!\cdots\!79}a^{15}-\frac{42\!\cdots\!19}{10\!\cdots\!59}a^{14}-\frac{10\!\cdots\!42}{10\!\cdots\!59}a^{13}+\frac{69\!\cdots\!39}{15\!\cdots\!37}a^{12}+\frac{55\!\cdots\!28}{10\!\cdots\!59}a^{11}+\frac{20\!\cdots\!57}{35\!\cdots\!53}a^{10}+\frac{35\!\cdots\!00}{16\!\cdots\!93}a^{9}-\frac{35\!\cdots\!82}{10\!\cdots\!59}a^{8}-\frac{36\!\cdots\!58}{35\!\cdots\!53}a^{7}-\frac{70\!\cdots\!54}{80\!\cdots\!43}a^{6}+\frac{78\!\cdots\!44}{10\!\cdots\!59}a^{5}+\frac{55\!\cdots\!30}{10\!\cdots\!59}a^{4}+\frac{42\!\cdots\!00}{10\!\cdots\!59}a^{3}+\frac{11\!\cdots\!40}{50\!\cdots\!79}a^{2}+\frac{46\!\cdots\!99}{35\!\cdots\!53}a+\frac{55\!\cdots\!06}{10\!\cdots\!59}$, $\frac{69\!\cdots\!46}{35\!\cdots\!53}a^{22}-\frac{34\!\cdots\!54}{35\!\cdots\!53}a^{21}+\frac{19\!\cdots\!28}{35\!\cdots\!53}a^{20}+\frac{15\!\cdots\!48}{35\!\cdots\!53}a^{19}+\frac{35\!\cdots\!99}{38\!\cdots\!17}a^{18}-\frac{81\!\cdots\!58}{11\!\cdots\!51}a^{17}-\frac{19\!\cdots\!08}{35\!\cdots\!53}a^{16}-\frac{18\!\cdots\!11}{11\!\cdots\!51}a^{15}-\frac{49\!\cdots\!55}{35\!\cdots\!53}a^{14}-\frac{55\!\cdots\!36}{35\!\cdots\!53}a^{13}+\frac{12\!\cdots\!66}{35\!\cdots\!53}a^{12}+\frac{12\!\cdots\!05}{35\!\cdots\!53}a^{11}+\frac{29\!\cdots\!87}{447800476377891}a^{10}+\frac{24\!\cdots\!64}{55\!\cdots\!31}a^{9}-\frac{18\!\cdots\!51}{35\!\cdots\!53}a^{8}-\frac{15\!\cdots\!97}{11\!\cdots\!51}a^{7}-\frac{82\!\cdots\!63}{930047143246389}a^{6}-\frac{15\!\cdots\!87}{35\!\cdots\!53}a^{5}+\frac{32\!\cdots\!49}{35\!\cdots\!53}a^{4}+\frac{28\!\cdots\!24}{35\!\cdots\!53}a^{3}+\frac{26\!\cdots\!59}{38\!\cdots\!17}a^{2}-\frac{22\!\cdots\!54}{11\!\cdots\!51}a+\frac{30\!\cdots\!95}{35\!\cdots\!53}$, $\frac{21\!\cdots\!46}{35\!\cdots\!53}a^{22}-\frac{17\!\cdots\!70}{50\!\cdots\!79}a^{21}+\frac{14\!\cdots\!98}{35\!\cdots\!53}a^{20}+\frac{407093290099495}{50\!\cdots\!79}a^{19}+\frac{92\!\cdots\!01}{38\!\cdots\!17}a^{18}-\frac{29\!\cdots\!57}{11\!\cdots\!51}a^{17}-\frac{52\!\cdots\!13}{35\!\cdots\!53}a^{16}-\frac{60\!\cdots\!45}{11\!\cdots\!51}a^{15}-\frac{18\!\cdots\!79}{35\!\cdots\!53}a^{14}-\frac{12\!\cdots\!60}{35\!\cdots\!53}a^{13}+\frac{37\!\cdots\!16}{35\!\cdots\!53}a^{12}+\frac{89\!\cdots\!27}{35\!\cdots\!53}a^{11}+\frac{16\!\cdots\!74}{55\!\cdots\!31}a^{10}+\frac{54\!\cdots\!47}{11\!\cdots\!51}a^{9}-\frac{39\!\cdots\!22}{35\!\cdots\!53}a^{8}-\frac{12\!\cdots\!70}{11\!\cdots\!51}a^{7}+\frac{930244317872363}{26\!\cdots\!81}a^{6}-\frac{12\!\cdots\!24}{35\!\cdots\!53}a^{5}+\frac{31\!\cdots\!81}{35\!\cdots\!53}a^{4}-\frac{35\!\cdots\!21}{50\!\cdots\!79}a^{3}+\frac{26\!\cdots\!06}{12\!\cdots\!39}a^{2}-\frac{15\!\cdots\!45}{18\!\cdots\!77}a+\frac{61\!\cdots\!10}{12\!\cdots\!57}$, $\frac{10\!\cdots\!00}{15\!\cdots\!37}a^{22}-\frac{31\!\cdots\!91}{10\!\cdots\!59}a^{21}+\frac{54\!\cdots\!80}{10\!\cdots\!59}a^{20}-\frac{11\!\cdots\!22}{10\!\cdots\!59}a^{19}+\frac{10\!\cdots\!42}{35\!\cdots\!53}a^{18}+\frac{12\!\cdots\!00}{50\!\cdots\!79}a^{17}+\frac{27\!\cdots\!27}{10\!\cdots\!59}a^{16}-\frac{12\!\cdots\!53}{50\!\cdots\!79}a^{15}-\frac{15\!\cdots\!05}{15\!\cdots\!37}a^{14}-\frac{15\!\cdots\!79}{10\!\cdots\!59}a^{13}-\frac{267649966888318}{51\!\cdots\!53}a^{12}+\frac{45\!\cdots\!34}{15\!\cdots\!37}a^{11}+\frac{94\!\cdots\!82}{35\!\cdots\!53}a^{10}+\frac{20\!\cdots\!88}{40\!\cdots\!19}a^{9}+\frac{30\!\cdots\!03}{10\!\cdots\!59}a^{8}-\frac{41\!\cdots\!31}{50\!\cdots\!79}a^{7}-\frac{24\!\cdots\!76}{80\!\cdots\!43}a^{6}-\frac{61\!\cdots\!87}{10\!\cdots\!59}a^{5}-\frac{16\!\cdots\!97}{10\!\cdots\!59}a^{4}+\frac{17\!\cdots\!04}{10\!\cdots\!59}a^{3}+\frac{11\!\cdots\!72}{35\!\cdots\!53}a^{2}+\frac{16\!\cdots\!66}{35\!\cdots\!53}a+\frac{10\!\cdots\!64}{10\!\cdots\!59}$, $\frac{81\!\cdots\!11}{10\!\cdots\!59}a^{22}-\frac{11\!\cdots\!74}{36\!\cdots\!71}a^{21}-\frac{41\!\cdots\!55}{10\!\cdots\!59}a^{20}+\frac{50\!\cdots\!84}{15\!\cdots\!37}a^{19}+\frac{12\!\cdots\!72}{35\!\cdots\!53}a^{18}+\frac{59\!\cdots\!14}{35\!\cdots\!53}a^{17}-\frac{21\!\cdots\!00}{10\!\cdots\!59}a^{16}-\frac{17\!\cdots\!61}{35\!\cdots\!53}a^{15}-\frac{55\!\cdots\!23}{10\!\cdots\!59}a^{14}-\frac{10\!\cdots\!46}{10\!\cdots\!59}a^{13}+\frac{24\!\cdots\!60}{10\!\cdots\!59}a^{12}+\frac{19\!\cdots\!12}{10\!\cdots\!59}a^{11}+\frac{10\!\cdots\!23}{35\!\cdots\!53}a^{10}+\frac{28\!\cdots\!71}{11\!\cdots\!51}a^{9}+\frac{17\!\cdots\!84}{10\!\cdots\!59}a^{8}-\frac{11\!\cdots\!00}{35\!\cdots\!53}a^{7}-\frac{38\!\cdots\!99}{80\!\cdots\!43}a^{6}-\frac{29\!\cdots\!31}{10\!\cdots\!59}a^{5}+\frac{12\!\cdots\!87}{10\!\cdots\!59}a^{4}+\frac{15\!\cdots\!55}{10\!\cdots\!59}a^{3}+\frac{41\!\cdots\!60}{12\!\cdots\!57}a^{2}+\frac{57\!\cdots\!77}{35\!\cdots\!53}a+\frac{43\!\cdots\!56}{10\!\cdots\!59}$, $\frac{10\!\cdots\!68}{10\!\cdots\!59}a^{22}-\frac{31\!\cdots\!92}{10\!\cdots\!59}a^{21}-\frac{62\!\cdots\!46}{10\!\cdots\!59}a^{20}+\frac{65\!\cdots\!66}{10\!\cdots\!59}a^{19}+\frac{17\!\cdots\!71}{35\!\cdots\!53}a^{18}+\frac{25\!\cdots\!75}{35\!\cdots\!53}a^{17}-\frac{31\!\cdots\!87}{10\!\cdots\!59}a^{16}-\frac{47\!\cdots\!65}{35\!\cdots\!53}a^{15}-\frac{30\!\cdots\!63}{17\!\cdots\!19}a^{14}-\frac{21\!\cdots\!53}{10\!\cdots\!59}a^{13}+\frac{90\!\cdots\!03}{10\!\cdots\!59}a^{12}+\frac{43\!\cdots\!78}{10\!\cdots\!59}a^{11}+\frac{24\!\cdots\!95}{35\!\cdots\!53}a^{10}+\frac{86\!\cdots\!35}{11\!\cdots\!51}a^{9}+\frac{16\!\cdots\!99}{10\!\cdots\!59}a^{8}-\frac{33\!\cdots\!90}{35\!\cdots\!53}a^{7}-\frac{11\!\cdots\!59}{80\!\cdots\!43}a^{6}-\frac{10\!\cdots\!20}{10\!\cdots\!59}a^{5}+\frac{15\!\cdots\!84}{10\!\cdots\!59}a^{4}+\frac{96\!\cdots\!87}{10\!\cdots\!59}a^{3}+\frac{48\!\cdots\!87}{50\!\cdots\!79}a^{2}+\frac{16\!\cdots\!19}{35\!\cdots\!53}a+\frac{12\!\cdots\!36}{10\!\cdots\!59}$, $\frac{73\!\cdots\!61}{10\!\cdots\!59}a^{22}-\frac{66\!\cdots\!11}{15\!\cdots\!37}a^{21}+\frac{69\!\cdots\!14}{10\!\cdots\!59}a^{20}+\frac{788745676440835}{36\!\cdots\!71}a^{19}+\frac{65\!\cdots\!09}{35\!\cdots\!53}a^{18}-\frac{14\!\cdots\!52}{35\!\cdots\!53}a^{17}-\frac{13\!\cdots\!27}{10\!\cdots\!59}a^{16}+\frac{16\!\cdots\!78}{35\!\cdots\!53}a^{15}-\frac{62\!\cdots\!05}{10\!\cdots\!59}a^{14}-\frac{10\!\cdots\!08}{10\!\cdots\!59}a^{13}+\frac{20\!\cdots\!70}{10\!\cdots\!59}a^{12}-\frac{15\!\cdots\!81}{10\!\cdots\!59}a^{11}-\frac{44\!\cdots\!92}{35\!\cdots\!53}a^{10}+\frac{72\!\cdots\!21}{11\!\cdots\!51}a^{9}-\frac{28\!\cdots\!73}{10\!\cdots\!59}a^{8}-\frac{62\!\cdots\!30}{35\!\cdots\!53}a^{7}+\frac{34\!\cdots\!06}{80\!\cdots\!43}a^{6}-\frac{16\!\cdots\!47}{10\!\cdots\!59}a^{5}-\frac{15\!\cdots\!87}{15\!\cdots\!37}a^{4}-\frac{11\!\cdots\!78}{10\!\cdots\!59}a^{3}+\frac{46\!\cdots\!71}{35\!\cdots\!53}a^{2}-\frac{65\!\cdots\!89}{50\!\cdots\!79}a+\frac{88\!\cdots\!84}{10\!\cdots\!59}$, $\frac{20\!\cdots\!48}{10\!\cdots\!59}a^{22}-\frac{15\!\cdots\!70}{15\!\cdots\!37}a^{21}+\frac{89\!\cdots\!32}{10\!\cdots\!59}a^{20}+\frac{47\!\cdots\!11}{10\!\cdots\!59}a^{19}+\frac{26\!\cdots\!82}{35\!\cdots\!53}a^{18}-\frac{10\!\cdots\!18}{35\!\cdots\!53}a^{17}-\frac{57\!\cdots\!88}{10\!\cdots\!59}a^{16}-\frac{22\!\cdots\!80}{35\!\cdots\!53}a^{15}-\frac{12\!\cdots\!56}{10\!\cdots\!59}a^{14}-\frac{16\!\cdots\!61}{10\!\cdots\!59}a^{13}+\frac{45\!\cdots\!16}{15\!\cdots\!37}a^{12}+\frac{25\!\cdots\!50}{10\!\cdots\!59}a^{11}+\frac{15\!\cdots\!07}{35\!\cdots\!53}a^{10}+\frac{46\!\cdots\!25}{11\!\cdots\!51}a^{9}-\frac{52\!\cdots\!76}{10\!\cdots\!59}a^{8}-\frac{26\!\cdots\!35}{35\!\cdots\!53}a^{7}-\frac{34\!\cdots\!40}{80\!\cdots\!43}a^{6}-\frac{67\!\cdots\!87}{10\!\cdots\!59}a^{5}+\frac{62\!\cdots\!39}{10\!\cdots\!59}a^{4}+\frac{29\!\cdots\!16}{10\!\cdots\!59}a^{3}+\frac{17\!\cdots\!66}{35\!\cdots\!53}a^{2}+\frac{84\!\cdots\!94}{35\!\cdots\!53}a+\frac{11\!\cdots\!73}{10\!\cdots\!59}$, $\frac{58127856312133}{38\!\cdots\!83}a^{22}-\frac{20\!\cdots\!95}{26\!\cdots\!81}a^{21}+\frac{655714741258427}{26\!\cdots\!81}a^{20}+\frac{34\!\cdots\!73}{26\!\cdots\!81}a^{19}+\frac{18\!\cdots\!98}{29\!\cdots\!09}a^{18}-\frac{14\!\cdots\!36}{89\!\cdots\!27}a^{17}-\frac{30\!\cdots\!25}{26\!\cdots\!81}a^{16}-\frac{892171523896219}{12\!\cdots\!61}a^{15}-\frac{67969719658520}{442153559904021}a^{14}-\frac{33\!\cdots\!20}{38\!\cdots\!83}a^{13}+\frac{63\!\cdots\!64}{26\!\cdots\!81}a^{12}+\frac{11\!\cdots\!30}{26\!\cdots\!81}a^{11}+\frac{805074382781338}{332979841409201}a^{10}+\frac{37\!\cdots\!89}{89\!\cdots\!27}a^{9}-\frac{14\!\cdots\!76}{26\!\cdots\!81}a^{8}-\frac{11\!\cdots\!94}{12\!\cdots\!61}a^{7}-\frac{70\!\cdots\!22}{26\!\cdots\!81}a^{6}+\frac{53\!\cdots\!13}{26\!\cdots\!81}a^{5}+\frac{15\!\cdots\!58}{26\!\cdots\!81}a^{4}+\frac{88\!\cdots\!25}{26\!\cdots\!81}a^{3}+\frac{722359281802572}{332979841409201}a^{2}-\frac{79271578972139}{428116938954687}a-\frac{31\!\cdots\!53}{26\!\cdots\!81}$, $a$
| 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: | \( 492297100.302 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 492297100.302 \cdot 1}{2\cdot\sqrt{69781355284066743169711357773860671}}\cr\approx \mathstrut & 1.12288719588 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 - 5*x^22 + 3*x^21 + 3*x^20 + 43*x^19 - 6*x^18 - 31*x^17 - 56*x^16 - 68*x^15 - 89*x^14 + 147*x^13 + 180*x^12 + 274*x^11 + 240*x^10 - 211*x^9 - 521*x^8 - 383*x^7 - 275*x^6 + 291*x^5 + 249*x^4 + 358*x^3 - 9*x^2 + 107*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^23 - 5*x^22 + 3*x^21 + 3*x^20 + 43*x^19 - 6*x^18 - 31*x^17 - 56*x^16 - 68*x^15 - 89*x^14 + 147*x^13 + 180*x^12 + 274*x^11 + 240*x^10 - 211*x^9 - 521*x^8 - 383*x^7 - 275*x^6 + 291*x^5 + 249*x^4 + 358*x^3 - 9*x^2 + 107*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^23 - 5*x^22 + 3*x^21 + 3*x^20 + 43*x^19 - 6*x^18 - 31*x^17 - 56*x^16 - 68*x^15 - 89*x^14 + 147*x^13 + 180*x^12 + 274*x^11 + 240*x^10 - 211*x^9 - 521*x^8 - 383*x^7 - 275*x^6 + 291*x^5 + 249*x^4 + 358*x^3 - 9*x^2 + 107*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^23 - 5*x^22 + 3*x^21 + 3*x^20 + 43*x^19 - 6*x^18 - 31*x^17 - 56*x^16 - 68*x^15 - 89*x^14 + 147*x^13 + 180*x^12 + 274*x^11 + 240*x^10 - 211*x^9 - 521*x^8 - 383*x^7 - 275*x^6 + 291*x^5 + 249*x^4 + 358*x^3 - 9*x^2 + 107*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
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 46 |
| The 13 conjugacy class representatives for $D_{23}$ |
| Character table for $D_{23}$ |
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
| Galois closure: | 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 | $23$ | ${\href{/padicField/3.2.0.1}{2} }^{11}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $23$ | ${\href{/padicField/7.2.0.1}{2} }^{11}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $23$ | ${\href{/padicField/13.2.0.1}{2} }^{11}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $23$ | $23$ | $23$ | ${\href{/padicField/29.2.0.1}{2} }^{11}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{11}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $23$ | $23$ | ${\href{/padicField/43.2.0.1}{2} }^{11}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{11}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $23$ | $23$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(1471\)
| $\Q_{1471}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |