Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 + 5*x^21 - 17*x^20 + 33*x^19 - 14*x^18 + 53*x^17 - 34*x^16 - 19*x^15 + 140*x^14 - 19*x^13 + 257*x^12 - 106*x^11 - 16*x^10 + 274*x^9 + 165*x^8 + 515*x^7 - 28*x^6 + 129*x^5 + 89*x^4 + 524*x^3 + 244*x^2 + 50*x - 1)
gp: K = bnfinit(y^23 + 5*y^21 - 17*y^20 + 33*y^19 - 14*y^18 + 53*y^17 - 34*y^16 - 19*y^15 + 140*y^14 - 19*y^13 + 257*y^12 - 106*y^11 - 16*y^10 + 274*y^9 + 165*y^8 + 515*y^7 - 28*y^6 + 129*y^5 + 89*y^4 + 524*y^3 + 244*y^2 + 50*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 + 5*x^21 - 17*x^20 + 33*x^19 - 14*x^18 + 53*x^17 - 34*x^16 - 19*x^15 + 140*x^14 - 19*x^13 + 257*x^12 - 106*x^11 - 16*x^10 + 274*x^9 + 165*x^8 + 515*x^7 - 28*x^6 + 129*x^5 + 89*x^4 + 524*x^3 + 244*x^2 + 50*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 + 5*x^21 - 17*x^20 + 33*x^19 - 14*x^18 + 53*x^17 - 34*x^16 - 19*x^15 + 140*x^14 - 19*x^13 + 257*x^12 - 106*x^11 - 16*x^10 + 274*x^9 + 165*x^8 + 515*x^7 - 28*x^6 + 129*x^5 + 89*x^4 + 524*x^3 + 244*x^2 + 50*x - 1)
\( x^{23} + 5 x^{21} - 17 x^{20} + 33 x^{19} - 14 x^{18} + 53 x^{17} - 34 x^{16} - 19 x^{15} + 140 x^{14} + \cdots - 1 \)
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: |
\(-2939886993377131174870795633320047\)
\(\medspace = -\,1103^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.52\) | 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: | $1103^{1/2}\approx 33.21144381083123$ | ||
| Ramified primes: |
\(1103\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1103}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5}a^{17}-\frac{1}{5}a^{16}-\frac{2}{5}a^{15}+\frac{1}{5}a^{14}-\frac{2}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{18}+\frac{2}{5}a^{16}-\frac{1}{5}a^{15}+\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{1}{5}a^{11}+\frac{2}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{35}a^{19}-\frac{3}{35}a^{18}+\frac{3}{35}a^{17}+\frac{1}{5}a^{16}-\frac{3}{35}a^{15}+\frac{11}{35}a^{14}+\frac{2}{35}a^{13}+\frac{2}{5}a^{12}-\frac{4}{35}a^{11}-\frac{1}{5}a^{10}-\frac{2}{35}a^{9}+\frac{4}{35}a^{8}-\frac{11}{35}a^{6}+\frac{1}{7}a^{5}-\frac{1}{5}a^{4}-\frac{2}{35}a^{3}+\frac{4}{35}a^{2}+\frac{1}{7}a-\frac{12}{35}$, $\frac{1}{665}a^{20}+\frac{3}{665}a^{19}-\frac{64}{665}a^{18}+\frac{18}{665}a^{17}-\frac{17}{665}a^{16}+\frac{43}{95}a^{15}+\frac{187}{665}a^{14}-\frac{331}{665}a^{13}+\frac{58}{133}a^{12}+\frac{137}{665}a^{11}+\frac{1}{35}a^{10}-\frac{239}{665}a^{9}+\frac{318}{665}a^{8}+\frac{276}{665}a^{7}+\frac{107}{665}a^{6}-\frac{243}{665}a^{5}+\frac{306}{665}a^{4}+\frac{258}{665}a^{3}-\frac{69}{665}a^{2}+\frac{12}{133}a+\frac{327}{665}$, $\frac{1}{23275}a^{21}+\frac{16}{23275}a^{20}-\frac{9}{3325}a^{19}+\frac{223}{3325}a^{18}-\frac{1759}{23275}a^{17}-\frac{452}{23275}a^{16}+\frac{48}{475}a^{15}+\frac{10061}{23275}a^{14}-\frac{10606}{23275}a^{13}-\frac{123}{4655}a^{12}+\frac{243}{931}a^{11}+\frac{8387}{23275}a^{10}-\frac{3644}{23275}a^{9}+\frac{6652}{23275}a^{8}-\frac{428}{23275}a^{7}+\frac{3694}{23275}a^{6}-\frac{3309}{23275}a^{5}-\frac{10793}{23275}a^{4}-\frac{363}{23275}a^{3}-\frac{5777}{23275}a^{2}-\frac{363}{3325}a-\frac{6199}{23275}$, $\frac{1}{30\!\cdots\!75}a^{22}+\frac{25\!\cdots\!48}{30\!\cdots\!75}a^{21}+\frac{55\!\cdots\!34}{30\!\cdots\!75}a^{20}-\frac{94\!\cdots\!56}{87\!\cdots\!25}a^{19}+\frac{25\!\cdots\!03}{30\!\cdots\!75}a^{18}+\frac{13\!\cdots\!16}{60\!\cdots\!75}a^{17}+\frac{77\!\cdots\!18}{30\!\cdots\!75}a^{16}-\frac{21\!\cdots\!64}{60\!\cdots\!75}a^{15}+\frac{32\!\cdots\!96}{30\!\cdots\!75}a^{14}+\frac{26\!\cdots\!98}{30\!\cdots\!75}a^{13}-\frac{32\!\cdots\!33}{60\!\cdots\!75}a^{12}-\frac{12\!\cdots\!38}{30\!\cdots\!75}a^{11}-\frac{24\!\cdots\!51}{60\!\cdots\!75}a^{10}+\frac{31\!\cdots\!94}{30\!\cdots\!75}a^{9}+\frac{85\!\cdots\!11}{30\!\cdots\!75}a^{8}+\frac{11\!\cdots\!18}{30\!\cdots\!75}a^{7}+\frac{25\!\cdots\!83}{23\!\cdots\!75}a^{6}+\frac{15\!\cdots\!27}{43\!\cdots\!25}a^{5}-\frac{38\!\cdots\!74}{30\!\cdots\!75}a^{4}-\frac{22\!\cdots\!63}{30\!\cdots\!75}a^{3}+\frac{28\!\cdots\!57}{12\!\cdots\!55}a^{2}+\frac{11\!\cdots\!69}{30\!\cdots\!75}a+\frac{73\!\cdots\!12}{30\!\cdots\!75}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
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: |
$\frac{38\!\cdots\!66}{42\!\cdots\!25}a^{22}-\frac{20\!\cdots\!47}{42\!\cdots\!25}a^{21}+\frac{19\!\cdots\!79}{42\!\cdots\!25}a^{20}-\frac{18\!\cdots\!74}{12\!\cdots\!75}a^{19}+\frac{13\!\cdots\!08}{42\!\cdots\!25}a^{18}-\frac{13\!\cdots\!62}{85\!\cdots\!25}a^{17}+\frac{12\!\cdots\!72}{22\!\cdots\!75}a^{16}-\frac{14\!\cdots\!39}{34\!\cdots\!81}a^{15}-\frac{81\!\cdots\!04}{42\!\cdots\!25}a^{14}+\frac{52\!\cdots\!08}{42\!\cdots\!25}a^{13}-\frac{82\!\cdots\!28}{85\!\cdots\!25}a^{12}+\frac{10\!\cdots\!92}{42\!\cdots\!25}a^{11}-\frac{10\!\cdots\!37}{85\!\cdots\!25}a^{10}+\frac{26\!\cdots\!14}{42\!\cdots\!25}a^{9}+\frac{93\!\cdots\!21}{42\!\cdots\!25}a^{8}+\frac{72\!\cdots\!58}{42\!\cdots\!25}a^{7}+\frac{17\!\cdots\!58}{33\!\cdots\!25}a^{6}-\frac{49\!\cdots\!63}{61\!\cdots\!75}a^{5}+\frac{78\!\cdots\!36}{42\!\cdots\!25}a^{4}+\frac{41\!\cdots\!37}{42\!\cdots\!25}a^{3}+\frac{47\!\cdots\!21}{85\!\cdots\!25}a^{2}+\frac{64\!\cdots\!44}{42\!\cdots\!25}a+\frac{36\!\cdots\!02}{42\!\cdots\!25}$, $\frac{21\!\cdots\!09}{60\!\cdots\!75}a^{22}+\frac{55\!\cdots\!44}{60\!\cdots\!75}a^{21}+\frac{19\!\cdots\!47}{12\!\cdots\!75}a^{20}-\frac{10\!\cdots\!58}{87\!\cdots\!25}a^{19}-\frac{29\!\cdots\!11}{60\!\cdots\!75}a^{18}+\frac{18\!\cdots\!12}{60\!\cdots\!75}a^{17}-\frac{44\!\cdots\!51}{87\!\cdots\!25}a^{16}+\frac{27\!\cdots\!04}{60\!\cdots\!75}a^{15}-\frac{28\!\cdots\!84}{60\!\cdots\!75}a^{14}+\frac{51\!\cdots\!51}{12\!\cdots\!55}a^{13}+\frac{86\!\cdots\!37}{64\!\cdots\!45}a^{12}+\frac{11\!\cdots\!73}{60\!\cdots\!75}a^{11}+\frac{14\!\cdots\!79}{60\!\cdots\!75}a^{10}-\frac{11\!\cdots\!12}{60\!\cdots\!75}a^{9}+\frac{10\!\cdots\!93}{60\!\cdots\!75}a^{8}+\frac{18\!\cdots\!51}{60\!\cdots\!75}a^{7}+\frac{11\!\cdots\!78}{46\!\cdots\!75}a^{6}+\frac{24\!\cdots\!48}{60\!\cdots\!75}a^{5}-\frac{49\!\cdots\!92}{60\!\cdots\!75}a^{4}+\frac{10\!\cdots\!27}{60\!\cdots\!75}a^{3}+\frac{20\!\cdots\!43}{87\!\cdots\!25}a^{2}+\frac{16\!\cdots\!26}{32\!\cdots\!25}a+\frac{37\!\cdots\!71}{91\!\cdots\!35}$, $\frac{40\!\cdots\!32}{22\!\cdots\!75}a^{22}+\frac{29\!\cdots\!33}{30\!\cdots\!75}a^{21}+\frac{26\!\cdots\!74}{30\!\cdots\!75}a^{20}-\frac{22\!\cdots\!67}{87\!\cdots\!25}a^{19}+\frac{17\!\cdots\!34}{43\!\cdots\!25}a^{18}+\frac{12\!\cdots\!31}{12\!\cdots\!55}a^{17}+\frac{22\!\cdots\!18}{30\!\cdots\!75}a^{16}-\frac{28\!\cdots\!83}{87\!\cdots\!25}a^{15}-\frac{24\!\cdots\!04}{30\!\cdots\!75}a^{14}+\frac{72\!\cdots\!43}{30\!\cdots\!75}a^{13}+\frac{57\!\cdots\!47}{60\!\cdots\!75}a^{12}+\frac{12\!\cdots\!47}{30\!\cdots\!75}a^{11}+\frac{38\!\cdots\!27}{60\!\cdots\!75}a^{10}-\frac{57\!\cdots\!91}{30\!\cdots\!75}a^{9}+\frac{14\!\cdots\!81}{30\!\cdots\!75}a^{8}+\frac{16\!\cdots\!23}{30\!\cdots\!75}a^{7}+\frac{24\!\cdots\!58}{23\!\cdots\!75}a^{6}+\frac{12\!\cdots\!04}{30\!\cdots\!75}a^{5}+\frac{33\!\cdots\!71}{30\!\cdots\!75}a^{4}+\frac{81\!\cdots\!87}{30\!\cdots\!75}a^{3}+\frac{58\!\cdots\!87}{60\!\cdots\!75}a^{2}+\frac{40\!\cdots\!17}{43\!\cdots\!25}a+\frac{58\!\cdots\!67}{30\!\cdots\!75}$, $\frac{43\!\cdots\!24}{60\!\cdots\!75}a^{22}-\frac{28\!\cdots\!14}{60\!\cdots\!75}a^{21}+\frac{69\!\cdots\!48}{12\!\cdots\!29}a^{20}-\frac{47\!\cdots\!93}{12\!\cdots\!75}a^{19}+\frac{68\!\cdots\!76}{60\!\cdots\!75}a^{18}-\frac{12\!\cdots\!01}{60\!\cdots\!75}a^{17}+\frac{12\!\cdots\!79}{60\!\cdots\!75}a^{16}-\frac{23\!\cdots\!67}{60\!\cdots\!75}a^{15}+\frac{21\!\cdots\!74}{87\!\cdots\!25}a^{14}+\frac{51\!\cdots\!83}{60\!\cdots\!75}a^{13}-\frac{93\!\cdots\!69}{12\!\cdots\!55}a^{12}+\frac{37\!\cdots\!68}{60\!\cdots\!75}a^{11}-\frac{15\!\cdots\!66}{87\!\cdots\!25}a^{10}+\frac{38\!\cdots\!73}{34\!\cdots\!93}a^{9}-\frac{12\!\cdots\!54}{87\!\cdots\!25}a^{8}-\frac{38\!\cdots\!46}{24\!\cdots\!51}a^{7}+\frac{28\!\cdots\!44}{46\!\cdots\!75}a^{6}-\frac{80\!\cdots\!99}{24\!\cdots\!51}a^{5}+\frac{67\!\cdots\!27}{60\!\cdots\!75}a^{4}-\frac{62\!\cdots\!04}{60\!\cdots\!75}a^{3}-\frac{67\!\cdots\!03}{60\!\cdots\!75}a^{2}-\frac{68\!\cdots\!63}{60\!\cdots\!75}a-\frac{21\!\cdots\!28}{60\!\cdots\!75}$, $\frac{55\!\cdots\!77}{30\!\cdots\!75}a^{22}+\frac{18\!\cdots\!51}{30\!\cdots\!75}a^{21}+\frac{34\!\cdots\!48}{30\!\cdots\!75}a^{20}+\frac{31\!\cdots\!97}{12\!\cdots\!75}a^{19}-\frac{11\!\cdots\!39}{30\!\cdots\!75}a^{18}+\frac{92\!\cdots\!33}{60\!\cdots\!75}a^{17}+\frac{49\!\cdots\!26}{30\!\cdots\!75}a^{16}+\frac{20\!\cdots\!39}{60\!\cdots\!75}a^{15}-\frac{70\!\cdots\!28}{30\!\cdots\!75}a^{14}+\frac{71\!\cdots\!66}{30\!\cdots\!75}a^{13}+\frac{25\!\cdots\!39}{60\!\cdots\!75}a^{12}+\frac{13\!\cdots\!49}{30\!\cdots\!75}a^{11}+\frac{94\!\cdots\!46}{64\!\cdots\!45}a^{10}-\frac{77\!\cdots\!82}{30\!\cdots\!75}a^{9}+\frac{21\!\cdots\!32}{30\!\cdots\!75}a^{8}+\frac{13\!\cdots\!21}{30\!\cdots\!75}a^{7}+\frac{28\!\cdots\!56}{23\!\cdots\!75}a^{6}+\frac{94\!\cdots\!08}{30\!\cdots\!75}a^{5}+\frac{58\!\cdots\!88}{62\!\cdots\!75}a^{4}+\frac{59\!\cdots\!84}{30\!\cdots\!75}a^{3}+\frac{93\!\cdots\!53}{60\!\cdots\!75}a^{2}+\frac{14\!\cdots\!58}{30\!\cdots\!75}a-\frac{22\!\cdots\!46}{30\!\cdots\!75}$, $\frac{34\!\cdots\!34}{30\!\cdots\!75}a^{22}+\frac{36\!\cdots\!67}{30\!\cdots\!75}a^{21}+\frac{24\!\cdots\!63}{43\!\cdots\!25}a^{20}-\frac{16\!\cdots\!17}{87\!\cdots\!25}a^{19}+\frac{11\!\cdots\!87}{30\!\cdots\!75}a^{18}-\frac{10\!\cdots\!79}{60\!\cdots\!75}a^{17}+\frac{26\!\cdots\!81}{43\!\cdots\!25}a^{16}-\frac{26\!\cdots\!27}{60\!\cdots\!75}a^{15}-\frac{60\!\cdots\!01}{30\!\cdots\!75}a^{14}+\frac{44\!\cdots\!72}{30\!\cdots\!75}a^{13}-\frac{14\!\cdots\!67}{60\!\cdots\!75}a^{12}+\frac{87\!\cdots\!33}{30\!\cdots\!75}a^{11}-\frac{80\!\cdots\!31}{64\!\cdots\!45}a^{10}-\frac{11\!\cdots\!19}{30\!\cdots\!75}a^{9}+\frac{84\!\cdots\!69}{30\!\cdots\!75}a^{8}+\frac{37\!\cdots\!57}{30\!\cdots\!75}a^{7}+\frac{13\!\cdots\!27}{23\!\cdots\!75}a^{6}-\frac{18\!\cdots\!14}{30\!\cdots\!75}a^{5}+\frac{36\!\cdots\!54}{30\!\cdots\!75}a^{4}+\frac{68\!\cdots\!53}{30\!\cdots\!75}a^{3}+\frac{36\!\cdots\!83}{87\!\cdots\!25}a^{2}+\frac{57\!\cdots\!61}{30\!\cdots\!75}a+\frac{12\!\cdots\!74}{43\!\cdots\!25}$, $\frac{70\!\cdots\!33}{30\!\cdots\!75}a^{22}+\frac{80\!\cdots\!74}{30\!\cdots\!75}a^{21}+\frac{18\!\cdots\!98}{16\!\cdots\!25}a^{20}-\frac{65\!\cdots\!33}{17\!\cdots\!65}a^{19}+\frac{21\!\cdots\!39}{30\!\cdots\!75}a^{18}-\frac{11\!\cdots\!14}{60\!\cdots\!75}a^{17}+\frac{33\!\cdots\!89}{30\!\cdots\!75}a^{16}-\frac{34\!\cdots\!66}{60\!\cdots\!75}a^{15}-\frac{25\!\cdots\!81}{43\!\cdots\!25}a^{14}+\frac{10\!\cdots\!19}{30\!\cdots\!75}a^{13}+\frac{52\!\cdots\!21}{60\!\cdots\!75}a^{12}+\frac{16\!\cdots\!71}{30\!\cdots\!75}a^{11}-\frac{15\!\cdots\!68}{12\!\cdots\!75}a^{10}-\frac{35\!\cdots\!19}{43\!\cdots\!25}a^{9}+\frac{31\!\cdots\!74}{43\!\cdots\!25}a^{8}+\frac{16\!\cdots\!74}{30\!\cdots\!75}a^{7}+\frac{28\!\cdots\!34}{23\!\cdots\!75}a^{6}+\frac{49\!\cdots\!77}{30\!\cdots\!75}a^{5}+\frac{89\!\cdots\!88}{30\!\cdots\!75}a^{4}+\frac{12\!\cdots\!76}{30\!\cdots\!75}a^{3}+\frac{80\!\cdots\!19}{60\!\cdots\!75}a^{2}+\frac{24\!\cdots\!62}{30\!\cdots\!75}a+\frac{32\!\cdots\!11}{30\!\cdots\!75}$, $\frac{53\!\cdots\!17}{30\!\cdots\!75}a^{22}-\frac{13\!\cdots\!99}{30\!\cdots\!75}a^{21}+\frac{27\!\cdots\!63}{30\!\cdots\!75}a^{20}-\frac{56\!\cdots\!67}{17\!\cdots\!65}a^{19}+\frac{20\!\cdots\!61}{30\!\cdots\!75}a^{18}-\frac{26\!\cdots\!91}{60\!\cdots\!75}a^{17}+\frac{33\!\cdots\!61}{30\!\cdots\!75}a^{16}-\frac{59\!\cdots\!04}{60\!\cdots\!75}a^{15}+\frac{38\!\cdots\!17}{30\!\cdots\!75}a^{14}+\frac{69\!\cdots\!56}{30\!\cdots\!75}a^{13}-\frac{41\!\cdots\!96}{60\!\cdots\!75}a^{12}+\frac{13\!\cdots\!04}{30\!\cdots\!75}a^{11}-\frac{91\!\cdots\!42}{32\!\cdots\!25}a^{10}+\frac{24\!\cdots\!58}{30\!\cdots\!75}a^{9}+\frac{14\!\cdots\!57}{30\!\cdots\!75}a^{8}+\frac{28\!\cdots\!54}{16\!\cdots\!25}a^{7}+\frac{21\!\cdots\!91}{23\!\cdots\!75}a^{6}-\frac{95\!\cdots\!52}{30\!\cdots\!75}a^{5}+\frac{12\!\cdots\!12}{30\!\cdots\!75}a^{4}+\frac{62\!\cdots\!26}{62\!\cdots\!75}a^{3}+\frac{57\!\cdots\!41}{60\!\cdots\!75}a^{2}+\frac{44\!\cdots\!77}{16\!\cdots\!25}a+\frac{19\!\cdots\!89}{30\!\cdots\!75}$, $\frac{10\!\cdots\!57}{30\!\cdots\!75}a^{22}-\frac{27\!\cdots\!59}{30\!\cdots\!75}a^{21}+\frac{70\!\cdots\!24}{43\!\cdots\!25}a^{20}-\frac{48\!\cdots\!91}{87\!\cdots\!25}a^{19}+\frac{35\!\cdots\!51}{30\!\cdots\!75}a^{18}-\frac{49\!\cdots\!67}{60\!\cdots\!75}a^{17}+\frac{27\!\cdots\!63}{43\!\cdots\!25}a^{16}-\frac{33\!\cdots\!91}{60\!\cdots\!75}a^{15}+\frac{10\!\cdots\!58}{16\!\cdots\!25}a^{14}+\frac{18\!\cdots\!31}{30\!\cdots\!75}a^{13}-\frac{15\!\cdots\!06}{60\!\cdots\!75}a^{12}+\frac{56\!\cdots\!34}{30\!\cdots\!75}a^{11}+\frac{60\!\cdots\!16}{12\!\cdots\!55}a^{10}+\frac{18\!\cdots\!63}{30\!\cdots\!75}a^{9}+\frac{18\!\cdots\!87}{30\!\cdots\!75}a^{8}-\frac{99\!\cdots\!14}{30\!\cdots\!75}a^{7}+\frac{14\!\cdots\!21}{23\!\cdots\!75}a^{6}+\frac{46\!\cdots\!28}{30\!\cdots\!75}a^{5}+\frac{53\!\cdots\!92}{30\!\cdots\!75}a^{4}-\frac{26\!\cdots\!56}{30\!\cdots\!75}a^{3}+\frac{55\!\cdots\!84}{87\!\cdots\!25}a^{2}+\frac{25\!\cdots\!53}{30\!\cdots\!75}a+\frac{29\!\cdots\!52}{43\!\cdots\!25}$, $\frac{12\!\cdots\!33}{30\!\cdots\!75}a^{22}-\frac{42\!\cdots\!86}{30\!\cdots\!75}a^{21}+\frac{64\!\cdots\!77}{30\!\cdots\!75}a^{20}-\frac{35\!\cdots\!33}{45\!\cdots\!75}a^{19}+\frac{49\!\cdots\!04}{30\!\cdots\!75}a^{18}-\frac{67\!\cdots\!56}{60\!\cdots\!75}a^{17}+\frac{77\!\cdots\!59}{30\!\cdots\!75}a^{16}-\frac{14\!\cdots\!69}{64\!\cdots\!45}a^{15}-\frac{43\!\cdots\!52}{30\!\cdots\!75}a^{14}+\frac{26\!\cdots\!47}{43\!\cdots\!25}a^{13}-\frac{24\!\cdots\!77}{87\!\cdots\!25}a^{12}+\frac{71\!\cdots\!79}{62\!\cdots\!75}a^{11}-\frac{50\!\cdots\!61}{60\!\cdots\!75}a^{10}+\frac{57\!\cdots\!32}{30\!\cdots\!75}a^{9}+\frac{35\!\cdots\!73}{30\!\cdots\!75}a^{8}+\frac{79\!\cdots\!29}{30\!\cdots\!75}a^{7}+\frac{35\!\cdots\!63}{17\!\cdots\!75}a^{6}-\frac{25\!\cdots\!83}{30\!\cdots\!75}a^{5}+\frac{22\!\cdots\!68}{30\!\cdots\!75}a^{4}+\frac{75\!\cdots\!81}{30\!\cdots\!75}a^{3}+\frac{12\!\cdots\!48}{60\!\cdots\!75}a^{2}+\frac{98\!\cdots\!22}{30\!\cdots\!75}a-\frac{91\!\cdots\!74}{30\!\cdots\!75}$, $\frac{30\!\cdots\!06}{30\!\cdots\!75}a^{22}+\frac{17\!\cdots\!77}{62\!\cdots\!75}a^{21}+\frac{15\!\cdots\!14}{30\!\cdots\!75}a^{20}-\frac{14\!\cdots\!94}{87\!\cdots\!25}a^{19}+\frac{98\!\cdots\!03}{30\!\cdots\!75}a^{18}-\frac{11\!\cdots\!66}{87\!\cdots\!25}a^{17}+\frac{15\!\cdots\!38}{30\!\cdots\!75}a^{16}-\frac{39\!\cdots\!76}{12\!\cdots\!55}a^{15}-\frac{60\!\cdots\!39}{30\!\cdots\!75}a^{14}+\frac{42\!\cdots\!53}{30\!\cdots\!75}a^{13}-\frac{88\!\cdots\!23}{60\!\cdots\!75}a^{12}+\frac{77\!\cdots\!97}{30\!\cdots\!75}a^{11}-\frac{60\!\cdots\!87}{60\!\cdots\!75}a^{10}-\frac{50\!\cdots\!26}{30\!\cdots\!75}a^{9}+\frac{82\!\cdots\!36}{30\!\cdots\!75}a^{8}+\frac{76\!\cdots\!29}{43\!\cdots\!25}a^{7}+\frac{12\!\cdots\!53}{23\!\cdots\!75}a^{6}-\frac{51\!\cdots\!06}{30\!\cdots\!75}a^{5}+\frac{40\!\cdots\!26}{30\!\cdots\!75}a^{4}+\frac{29\!\cdots\!92}{30\!\cdots\!75}a^{3}+\frac{32\!\cdots\!46}{60\!\cdots\!75}a^{2}+\frac{78\!\cdots\!29}{30\!\cdots\!75}a+\frac{16\!\cdots\!32}{30\!\cdots\!75}$
| 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: | \( 42306743.1121 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 42306743.1121 \cdot 1}{2\cdot\sqrt{2939886993377131174870795633320047}}\cr\approx \mathstrut & 0.470135262821 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 + 5*x^21 - 17*x^20 + 33*x^19 - 14*x^18 + 53*x^17 - 34*x^16 - 19*x^15 + 140*x^14 - 19*x^13 + 257*x^12 - 106*x^11 - 16*x^10 + 274*x^9 + 165*x^8 + 515*x^7 - 28*x^6 + 129*x^5 + 89*x^4 + 524*x^3 + 244*x^2 + 50*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^21 - 17*x^20 + 33*x^19 - 14*x^18 + 53*x^17 - 34*x^16 - 19*x^15 + 140*x^14 - 19*x^13 + 257*x^12 - 106*x^11 - 16*x^10 + 274*x^9 + 165*x^8 + 515*x^7 - 28*x^6 + 129*x^5 + 89*x^4 + 524*x^3 + 244*x^2 + 50*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^21 - 17*x^20 + 33*x^19 - 14*x^18 + 53*x^17 - 34*x^16 - 19*x^15 + 140*x^14 - 19*x^13 + 257*x^12 - 106*x^11 - 16*x^10 + 274*x^9 + 165*x^8 + 515*x^7 - 28*x^6 + 129*x^5 + 89*x^4 + 524*x^3 + 244*x^2 + 50*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^21 - 17*x^20 + 33*x^19 - 14*x^18 + 53*x^17 - 34*x^16 - 19*x^15 + 140*x^14 - 19*x^13 + 257*x^12 - 106*x^11 - 16*x^10 + 274*x^9 + 165*x^8 + 515*x^7 - 28*x^6 + 129*x^5 + 89*x^4 + 524*x^3 + 244*x^2 + 50*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: | deg 46 |
| 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 | $23$ | $23$ | ${\href{/padicField/5.2.0.1}{2} }^{11}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.2.0.1}{2} }^{11}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{11}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{11}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $23$ | ${\href{/padicField/19.2.0.1}{2} }^{11}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $23$ | $23$ | ${\href{/padicField/31.2.0.1}{2} }^{11}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $23$ | $23$ | $23$ | $23$ | $23$ | ${\href{/padicField/59.2.0.1}{2} }^{11}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(1103\)
| $\Q_{1103}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |