Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 9*x^25 + 17*x^24 - 6*x^23 - 86*x^22 + 42*x^21 - 26*x^20 + 2*x^19 + 222*x^18 - 620*x^17 - 136*x^16 + 1722*x^15 - 1022*x^14 - 1740*x^13 + 3140*x^12 + 278*x^11 - 3510*x^10 + 1746*x^9 + 1702*x^8 - 1886*x^7 - 6*x^6 + 1068*x^5 - 272*x^4 - 230*x^3 + 166*x^2 + 10*x - 58)
gp: K = bnfinit(y^27 - 3*y^26 - 9*y^25 + 17*y^24 - 6*y^23 - 86*y^22 + 42*y^21 - 26*y^20 + 2*y^19 + 222*y^18 - 620*y^17 - 136*y^16 + 1722*y^15 - 1022*y^14 - 1740*y^13 + 3140*y^12 + 278*y^11 - 3510*y^10 + 1746*y^9 + 1702*y^8 - 1886*y^7 - 6*y^6 + 1068*y^5 - 272*y^4 - 230*y^3 + 166*y^2 + 10*y - 58, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 3*x^26 - 9*x^25 + 17*x^24 - 6*x^23 - 86*x^22 + 42*x^21 - 26*x^20 + 2*x^19 + 222*x^18 - 620*x^17 - 136*x^16 + 1722*x^15 - 1022*x^14 - 1740*x^13 + 3140*x^12 + 278*x^11 - 3510*x^10 + 1746*x^9 + 1702*x^8 - 1886*x^7 - 6*x^6 + 1068*x^5 - 272*x^4 - 230*x^3 + 166*x^2 + 10*x - 58);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 3*x^26 - 9*x^25 + 17*x^24 - 6*x^23 - 86*x^22 + 42*x^21 - 26*x^20 + 2*x^19 + 222*x^18 - 620*x^17 - 136*x^16 + 1722*x^15 - 1022*x^14 - 1740*x^13 + 3140*x^12 + 278*x^11 - 3510*x^10 + 1746*x^9 + 1702*x^8 - 1886*x^7 - 6*x^6 + 1068*x^5 - 272*x^4 - 230*x^3 + 166*x^2 + 10*x - 58)
\( x^{27} - 3 x^{26} - 9 x^{25} + 17 x^{24} - 6 x^{23} - 86 x^{22} + 42 x^{21} - 26 x^{20} + 2 x^{19} + \cdots - 58 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4852102490441335701504000000000000000000\)
\(\medspace = 2^{52}\cdot 3^{24}\cdot 5^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.50\) | 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: | $2^{25/12}3^{7/6}5^{3/4}\approx 51.05223872742641$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{5}a^{22}+\frac{2}{5}a^{21}+\frac{1}{5}a^{20}+\frac{2}{5}a^{19}-\frac{1}{5}a^{18}-\frac{2}{5}a^{17}-\frac{2}{5}a^{15}-\frac{2}{5}a^{13}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{2}+\frac{2}{5}$, $\frac{1}{5}a^{23}+\frac{2}{5}a^{21}-\frac{1}{5}a^{17}-\frac{2}{5}a^{16}-\frac{1}{5}a^{15}-\frac{2}{5}a^{14}+\frac{1}{5}a^{13}+\frac{1}{5}a^{11}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{24}+\frac{1}{5}a^{21}-\frac{2}{5}a^{20}+\frac{1}{5}a^{19}+\frac{1}{5}a^{18}+\frac{2}{5}a^{17}-\frac{1}{5}a^{16}+\frac{2}{5}a^{15}+\frac{1}{5}a^{14}-\frac{1}{5}a^{13}+\frac{2}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{3}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{25}+\frac{1}{5}a^{21}-\frac{1}{5}a^{19}-\frac{2}{5}a^{18}+\frac{1}{5}a^{17}+\frac{2}{5}a^{16}-\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{1}{5}a^{13}-\frac{1}{5}a^{11}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{11\!\cdots\!75}a^{26}-\frac{97\!\cdots\!64}{11\!\cdots\!75}a^{25}+\frac{68\!\cdots\!03}{44\!\cdots\!83}a^{24}-\frac{61\!\cdots\!43}{11\!\cdots\!75}a^{23}-\frac{10\!\cdots\!53}{11\!\cdots\!75}a^{22}+\frac{48\!\cdots\!07}{11\!\cdots\!75}a^{21}-\frac{37\!\cdots\!17}{22\!\cdots\!15}a^{20}+\frac{45\!\cdots\!14}{11\!\cdots\!75}a^{19}+\frac{54\!\cdots\!24}{23\!\cdots\!25}a^{18}+\frac{55\!\cdots\!59}{11\!\cdots\!75}a^{17}-\frac{45\!\cdots\!34}{11\!\cdots\!75}a^{16}+\frac{15\!\cdots\!48}{11\!\cdots\!75}a^{15}-\frac{40\!\cdots\!11}{11\!\cdots\!75}a^{14}+\frac{27\!\cdots\!49}{11\!\cdots\!75}a^{13}+\frac{45\!\cdots\!11}{11\!\cdots\!75}a^{12}+\frac{30\!\cdots\!94}{11\!\cdots\!75}a^{11}+\frac{98\!\cdots\!64}{11\!\cdots\!75}a^{10}+\frac{51\!\cdots\!66}{11\!\cdots\!75}a^{9}-\frac{34\!\cdots\!69}{22\!\cdots\!15}a^{8}+\frac{51\!\cdots\!47}{11\!\cdots\!75}a^{7}-\frac{47\!\cdots\!03}{11\!\cdots\!75}a^{6}+\frac{35\!\cdots\!17}{11\!\cdots\!75}a^{5}+\frac{30\!\cdots\!96}{11\!\cdots\!75}a^{4}-\frac{21\!\cdots\!03}{11\!\cdots\!75}a^{3}+\frac{11\!\cdots\!24}{23\!\cdots\!25}a^{2}-\frac{31\!\cdots\!17}{11\!\cdots\!75}a-\frac{55\!\cdots\!73}{11\!\cdots\!75}$
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
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: | $14$ | 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: |
$\frac{35\!\cdots\!57}{11\!\cdots\!75}a^{26}+\frac{28\!\cdots\!62}{11\!\cdots\!75}a^{25}-\frac{13\!\cdots\!64}{22\!\cdots\!15}a^{24}-\frac{82\!\cdots\!46}{11\!\cdots\!75}a^{23}+\frac{16\!\cdots\!54}{11\!\cdots\!75}a^{22}-\frac{26\!\cdots\!76}{11\!\cdots\!75}a^{21}-\frac{21\!\cdots\!51}{22\!\cdots\!15}a^{20}-\frac{69\!\cdots\!07}{11\!\cdots\!75}a^{19}+\frac{16\!\cdots\!48}{23\!\cdots\!25}a^{18}+\frac{77\!\cdots\!28}{11\!\cdots\!75}a^{17}+\frac{85\!\cdots\!42}{11\!\cdots\!75}a^{16}-\frac{77\!\cdots\!04}{11\!\cdots\!75}a^{15}-\frac{60\!\cdots\!42}{11\!\cdots\!75}a^{14}+\frac{19\!\cdots\!58}{11\!\cdots\!75}a^{13}-\frac{80\!\cdots\!88}{11\!\cdots\!75}a^{12}-\frac{21\!\cdots\!12}{11\!\cdots\!75}a^{11}+\frac{31\!\cdots\!03}{11\!\cdots\!75}a^{10}+\frac{14\!\cdots\!92}{11\!\cdots\!75}a^{9}-\frac{15\!\cdots\!79}{44\!\cdots\!83}a^{8}+\frac{36\!\cdots\!29}{11\!\cdots\!75}a^{7}+\frac{28\!\cdots\!14}{11\!\cdots\!75}a^{6}-\frac{99\!\cdots\!01}{11\!\cdots\!75}a^{5}-\frac{78\!\cdots\!33}{11\!\cdots\!75}a^{4}+\frac{99\!\cdots\!74}{11\!\cdots\!75}a^{3}+\frac{47\!\cdots\!38}{23\!\cdots\!25}a^{2}-\frac{25\!\cdots\!44}{11\!\cdots\!75}a-\frac{29\!\cdots\!21}{11\!\cdots\!75}$, $\frac{27\!\cdots\!14}{11\!\cdots\!75}a^{26}+\frac{74\!\cdots\!31}{11\!\cdots\!75}a^{25}+\frac{11\!\cdots\!72}{44\!\cdots\!83}a^{24}-\frac{40\!\cdots\!03}{11\!\cdots\!75}a^{23}-\frac{81\!\cdots\!03}{11\!\cdots\!75}a^{22}+\frac{24\!\cdots\!12}{11\!\cdots\!75}a^{21}-\frac{75\!\cdots\!16}{22\!\cdots\!15}a^{20}-\frac{36\!\cdots\!21}{11\!\cdots\!75}a^{19}-\frac{90\!\cdots\!61}{23\!\cdots\!25}a^{18}-\frac{64\!\cdots\!46}{11\!\cdots\!75}a^{17}+\frac{15\!\cdots\!56}{11\!\cdots\!75}a^{16}+\frac{10\!\cdots\!78}{11\!\cdots\!75}a^{15}-\frac{49\!\cdots\!21}{11\!\cdots\!75}a^{14}+\frac{71\!\cdots\!39}{11\!\cdots\!75}a^{13}+\frac{65\!\cdots\!06}{11\!\cdots\!75}a^{12}-\frac{63\!\cdots\!61}{11\!\cdots\!75}a^{11}-\frac{48\!\cdots\!11}{11\!\cdots\!75}a^{10}+\frac{96\!\cdots\!46}{11\!\cdots\!75}a^{9}+\frac{51\!\cdots\!84}{22\!\cdots\!15}a^{8}-\frac{67\!\cdots\!58}{11\!\cdots\!75}a^{7}+\frac{22\!\cdots\!52}{11\!\cdots\!75}a^{6}+\frac{24\!\cdots\!77}{11\!\cdots\!75}a^{5}-\frac{18\!\cdots\!99}{11\!\cdots\!75}a^{4}-\frac{27\!\cdots\!13}{11\!\cdots\!75}a^{3}+\frac{14\!\cdots\!59}{23\!\cdots\!25}a^{2}+\frac{39\!\cdots\!63}{11\!\cdots\!75}a-\frac{98\!\cdots\!43}{11\!\cdots\!75}$, $\frac{86\!\cdots\!29}{11\!\cdots\!75}a^{26}+\frac{22\!\cdots\!76}{11\!\cdots\!75}a^{25}+\frac{17\!\cdots\!66}{22\!\cdots\!15}a^{24}-\frac{12\!\cdots\!93}{11\!\cdots\!75}a^{23}+\frac{90\!\cdots\!47}{11\!\cdots\!75}a^{22}+\frac{77\!\cdots\!12}{11\!\cdots\!75}a^{21}-\frac{38\!\cdots\!32}{22\!\cdots\!15}a^{20}+\frac{26\!\cdots\!49}{11\!\cdots\!75}a^{19}+\frac{41\!\cdots\!94}{23\!\cdots\!25}a^{18}-\frac{27\!\cdots\!61}{11\!\cdots\!75}a^{17}+\frac{56\!\cdots\!26}{11\!\cdots\!75}a^{16}+\frac{26\!\cdots\!98}{11\!\cdots\!75}a^{15}-\frac{14\!\cdots\!41}{11\!\cdots\!75}a^{14}+\frac{66\!\cdots\!44}{11\!\cdots\!75}a^{13}+\frac{12\!\cdots\!11}{11\!\cdots\!75}a^{12}-\frac{23\!\cdots\!36}{11\!\cdots\!75}a^{11}+\frac{29\!\cdots\!54}{11\!\cdots\!75}a^{10}+\frac{18\!\cdots\!11}{11\!\cdots\!75}a^{9}-\frac{33\!\cdots\!18}{22\!\cdots\!15}a^{8}+\frac{48\!\cdots\!82}{11\!\cdots\!75}a^{7}+\frac{45\!\cdots\!32}{11\!\cdots\!75}a^{6}-\frac{16\!\cdots\!33}{11\!\cdots\!75}a^{5}+\frac{73\!\cdots\!11}{11\!\cdots\!75}a^{4}+\frac{12\!\cdots\!12}{11\!\cdots\!75}a^{3}-\frac{10\!\cdots\!26}{23\!\cdots\!25}a^{2}+\frac{15\!\cdots\!88}{11\!\cdots\!75}a+\frac{12\!\cdots\!37}{11\!\cdots\!75}$, $\frac{21\!\cdots\!72}{11\!\cdots\!75}a^{26}-\frac{52\!\cdots\!78}{11\!\cdots\!75}a^{25}-\frac{46\!\cdots\!97}{22\!\cdots\!15}a^{24}+\frac{27\!\cdots\!29}{11\!\cdots\!75}a^{23}+\frac{14\!\cdots\!49}{11\!\cdots\!75}a^{22}-\frac{19\!\cdots\!21}{11\!\cdots\!75}a^{21}-\frac{20\!\cdots\!42}{22\!\cdots\!15}a^{20}+\frac{52\!\cdots\!48}{11\!\cdots\!75}a^{19}+\frac{24\!\cdots\!23}{23\!\cdots\!25}a^{18}+\frac{49\!\cdots\!28}{11\!\cdots\!75}a^{17}-\frac{10\!\cdots\!78}{11\!\cdots\!75}a^{16}-\frac{11\!\cdots\!29}{11\!\cdots\!75}a^{15}+\frac{38\!\cdots\!93}{11\!\cdots\!75}a^{14}+\frac{17\!\cdots\!03}{11\!\cdots\!75}a^{13}-\frac{58\!\cdots\!23}{11\!\cdots\!75}a^{12}+\frac{43\!\cdots\!03}{11\!\cdots\!75}a^{11}+\frac{53\!\cdots\!83}{11\!\cdots\!75}a^{10}-\frac{78\!\cdots\!03}{11\!\cdots\!75}a^{9}-\frac{73\!\cdots\!22}{44\!\cdots\!83}a^{8}+\frac{66\!\cdots\!59}{11\!\cdots\!75}a^{7}-\frac{11\!\cdots\!71}{11\!\cdots\!75}a^{6}-\frac{30\!\cdots\!11}{11\!\cdots\!75}a^{5}+\frac{17\!\cdots\!07}{11\!\cdots\!75}a^{4}+\frac{10\!\cdots\!79}{11\!\cdots\!75}a^{3}-\frac{15\!\cdots\!77}{23\!\cdots\!25}a^{2}-\frac{10\!\cdots\!04}{11\!\cdots\!75}a+\frac{79\!\cdots\!79}{11\!\cdots\!75}$, $\frac{17\!\cdots\!08}{11\!\cdots\!75}a^{26}-\frac{35\!\cdots\!72}{11\!\cdots\!75}a^{25}-\frac{42\!\cdots\!03}{22\!\cdots\!15}a^{24}+\frac{13\!\cdots\!26}{11\!\cdots\!75}a^{23}+\frac{20\!\cdots\!06}{11\!\cdots\!75}a^{22}-\frac{15\!\cdots\!94}{11\!\cdots\!75}a^{21}-\frac{17\!\cdots\!99}{22\!\cdots\!15}a^{20}+\frac{10\!\cdots\!17}{11\!\cdots\!75}a^{19}+\frac{85\!\cdots\!17}{23\!\cdots\!25}a^{18}+\frac{41\!\cdots\!02}{11\!\cdots\!75}a^{17}-\frac{71\!\cdots\!17}{11\!\cdots\!75}a^{16}-\frac{13\!\cdots\!31}{11\!\cdots\!75}a^{15}+\frac{25\!\cdots\!67}{11\!\cdots\!75}a^{14}+\frac{16\!\cdots\!52}{11\!\cdots\!75}a^{13}-\frac{41\!\cdots\!82}{11\!\cdots\!75}a^{12}+\frac{98\!\cdots\!72}{11\!\cdots\!75}a^{11}+\frac{55\!\cdots\!47}{11\!\cdots\!75}a^{10}-\frac{33\!\cdots\!67}{11\!\cdots\!75}a^{9}-\frac{83\!\cdots\!36}{22\!\cdots\!15}a^{8}+\frac{36\!\cdots\!46}{11\!\cdots\!75}a^{7}+\frac{18\!\cdots\!56}{11\!\cdots\!75}a^{6}-\frac{19\!\cdots\!59}{11\!\cdots\!75}a^{5}+\frac{12\!\cdots\!38}{11\!\cdots\!75}a^{4}+\frac{93\!\cdots\!01}{11\!\cdots\!75}a^{3}-\frac{39\!\cdots\!63}{23\!\cdots\!25}a^{2}-\frac{23\!\cdots\!46}{11\!\cdots\!75}a+\frac{67\!\cdots\!01}{11\!\cdots\!75}$, $\frac{99\!\cdots\!71}{11\!\cdots\!75}a^{26}-\frac{99\!\cdots\!91}{11\!\cdots\!75}a^{25}+\frac{10\!\cdots\!04}{22\!\cdots\!15}a^{24}+\frac{83\!\cdots\!13}{11\!\cdots\!75}a^{23}-\frac{22\!\cdots\!42}{11\!\cdots\!75}a^{22}+\frac{29\!\cdots\!23}{11\!\cdots\!75}a^{21}+\frac{19\!\cdots\!28}{22\!\cdots\!15}a^{20}-\frac{71\!\cdots\!74}{11\!\cdots\!75}a^{19}+\frac{23\!\cdots\!31}{23\!\cdots\!25}a^{18}-\frac{34\!\cdots\!84}{11\!\cdots\!75}a^{17}-\frac{16\!\cdots\!46}{11\!\cdots\!75}a^{16}+\frac{88\!\cdots\!27}{11\!\cdots\!75}a^{15}-\frac{35\!\cdots\!59}{11\!\cdots\!75}a^{14}-\frac{17\!\cdots\!69}{11\!\cdots\!75}a^{13}+\frac{21\!\cdots\!74}{11\!\cdots\!75}a^{12}+\frac{62\!\cdots\!16}{11\!\cdots\!75}a^{11}-\frac{42\!\cdots\!19}{11\!\cdots\!75}a^{10}+\frac{16\!\cdots\!59}{11\!\cdots\!75}a^{9}+\frac{53\!\cdots\!51}{22\!\cdots\!15}a^{8}-\frac{36\!\cdots\!32}{11\!\cdots\!75}a^{7}-\frac{11\!\cdots\!97}{11\!\cdots\!75}a^{6}+\frac{20\!\cdots\!73}{11\!\cdots\!75}a^{5}-\frac{78\!\cdots\!16}{11\!\cdots\!75}a^{4}-\frac{53\!\cdots\!92}{11\!\cdots\!75}a^{3}+\frac{10\!\cdots\!46}{23\!\cdots\!25}a^{2}+\frac{23\!\cdots\!37}{11\!\cdots\!75}a-\frac{13\!\cdots\!27}{11\!\cdots\!75}$, $\frac{50\!\cdots\!59}{22\!\cdots\!15}a^{26}+\frac{16\!\cdots\!56}{22\!\cdots\!15}a^{25}+\frac{38\!\cdots\!62}{22\!\cdots\!15}a^{24}-\frac{92\!\cdots\!38}{22\!\cdots\!15}a^{23}+\frac{67\!\cdots\!24}{22\!\cdots\!15}a^{22}+\frac{38\!\cdots\!43}{22\!\cdots\!15}a^{21}-\frac{30\!\cdots\!87}{22\!\cdots\!15}a^{20}+\frac{64\!\cdots\!65}{44\!\cdots\!83}a^{19}-\frac{45\!\cdots\!56}{47\!\cdots\!45}a^{18}-\frac{95\!\cdots\!41}{22\!\cdots\!15}a^{17}+\frac{33\!\cdots\!89}{22\!\cdots\!15}a^{16}-\frac{59\!\cdots\!77}{22\!\cdots\!15}a^{15}-\frac{75\!\cdots\!59}{22\!\cdots\!15}a^{14}+\frac{71\!\cdots\!28}{22\!\cdots\!15}a^{13}+\frac{45\!\cdots\!54}{22\!\cdots\!15}a^{12}-\frac{14\!\cdots\!24}{22\!\cdots\!15}a^{11}+\frac{40\!\cdots\!23}{22\!\cdots\!15}a^{10}+\frac{11\!\cdots\!83}{22\!\cdots\!15}a^{9}-\frac{10\!\cdots\!38}{22\!\cdots\!15}a^{8}-\frac{16\!\cdots\!58}{22\!\cdots\!15}a^{7}+\frac{55\!\cdots\!41}{22\!\cdots\!15}a^{6}-\frac{21\!\cdots\!22}{22\!\cdots\!15}a^{5}-\frac{15\!\cdots\!18}{22\!\cdots\!15}a^{4}+\frac{65\!\cdots\!38}{22\!\cdots\!15}a^{3}-\frac{16\!\cdots\!95}{94\!\cdots\!89}a^{2}-\frac{41\!\cdots\!60}{44\!\cdots\!83}a-\frac{57\!\cdots\!97}{22\!\cdots\!15}$, $\frac{29\!\cdots\!57}{11\!\cdots\!75}a^{26}+\frac{90\!\cdots\!88}{11\!\cdots\!75}a^{25}+\frac{50\!\cdots\!24}{22\!\cdots\!15}a^{24}-\frac{49\!\cdots\!49}{11\!\cdots\!75}a^{23}+\frac{26\!\cdots\!31}{11\!\cdots\!75}a^{22}+\frac{23\!\cdots\!06}{11\!\cdots\!75}a^{21}-\frac{25\!\cdots\!32}{22\!\cdots\!15}a^{20}+\frac{14\!\cdots\!52}{11\!\cdots\!75}a^{19}-\frac{20\!\cdots\!28}{23\!\cdots\!25}a^{18}-\frac{60\!\cdots\!28}{11\!\cdots\!75}a^{17}+\frac{18\!\cdots\!98}{11\!\cdots\!75}a^{16}+\frac{93\!\cdots\!29}{11\!\cdots\!75}a^{15}-\frac{45\!\cdots\!18}{11\!\cdots\!75}a^{14}+\frac{31\!\cdots\!02}{11\!\cdots\!75}a^{13}+\frac{35\!\cdots\!33}{11\!\cdots\!75}a^{12}-\frac{78\!\cdots\!68}{11\!\cdots\!75}a^{11}+\frac{51\!\cdots\!62}{11\!\cdots\!75}a^{10}+\frac{70\!\cdots\!18}{11\!\cdots\!75}a^{9}-\frac{86\!\cdots\!57}{22\!\cdots\!15}a^{8}-\frac{19\!\cdots\!59}{11\!\cdots\!75}a^{7}+\frac{29\!\cdots\!91}{11\!\cdots\!75}a^{6}-\frac{64\!\cdots\!84}{11\!\cdots\!75}a^{5}-\frac{10\!\cdots\!92}{11\!\cdots\!75}a^{4}+\frac{31\!\cdots\!56}{11\!\cdots\!75}a^{3}-\frac{16\!\cdots\!38}{23\!\cdots\!25}a^{2}-\frac{69\!\cdots\!96}{11\!\cdots\!75}a+\frac{96\!\cdots\!71}{11\!\cdots\!75}$, $\frac{21\!\cdots\!72}{22\!\cdots\!15}a^{26}+\frac{11\!\cdots\!93}{22\!\cdots\!15}a^{25}+\frac{11\!\cdots\!83}{22\!\cdots\!15}a^{24}-\frac{83\!\cdots\!88}{22\!\cdots\!15}a^{23}+\frac{11\!\cdots\!78}{22\!\cdots\!15}a^{22}+\frac{26\!\cdots\!61}{44\!\cdots\!83}a^{21}-\frac{56\!\cdots\!14}{22\!\cdots\!15}a^{20}+\frac{38\!\cdots\!19}{22\!\cdots\!15}a^{19}-\frac{82\!\cdots\!46}{94\!\cdots\!89}a^{18}-\frac{41\!\cdots\!77}{22\!\cdots\!15}a^{17}+\frac{26\!\cdots\!48}{22\!\cdots\!15}a^{16}-\frac{70\!\cdots\!99}{44\!\cdots\!83}a^{15}-\frac{37\!\cdots\!07}{22\!\cdots\!15}a^{14}+\frac{11\!\cdots\!11}{22\!\cdots\!15}a^{13}-\frac{39\!\cdots\!97}{22\!\cdots\!15}a^{12}-\frac{14\!\cdots\!68}{22\!\cdots\!15}a^{11}+\frac{18\!\cdots\!14}{22\!\cdots\!15}a^{10}+\frac{43\!\cdots\!14}{22\!\cdots\!15}a^{9}-\frac{43\!\cdots\!24}{44\!\cdots\!83}a^{8}+\frac{10\!\cdots\!21}{22\!\cdots\!15}a^{7}+\frac{87\!\cdots\!56}{22\!\cdots\!15}a^{6}-\frac{21\!\cdots\!81}{44\!\cdots\!83}a^{5}+\frac{17\!\cdots\!08}{22\!\cdots\!15}a^{4}+\frac{37\!\cdots\!76}{22\!\cdots\!15}a^{3}-\frac{41\!\cdots\!58}{47\!\cdots\!45}a^{2}-\frac{47\!\cdots\!81}{22\!\cdots\!15}a+\frac{51\!\cdots\!19}{22\!\cdots\!15}$, $\frac{54\!\cdots\!46}{22\!\cdots\!15}a^{26}+\frac{16\!\cdots\!78}{22\!\cdots\!15}a^{25}+\frac{99\!\cdots\!61}{44\!\cdots\!83}a^{24}-\frac{10\!\cdots\!32}{22\!\cdots\!15}a^{23}+\frac{19\!\cdots\!58}{22\!\cdots\!15}a^{22}+\frac{48\!\cdots\!32}{22\!\cdots\!15}a^{21}-\frac{51\!\cdots\!75}{44\!\cdots\!83}a^{20}+\frac{12\!\cdots\!22}{22\!\cdots\!15}a^{19}-\frac{39\!\cdots\!88}{47\!\cdots\!45}a^{18}-\frac{25\!\cdots\!80}{44\!\cdots\!83}a^{17}+\frac{35\!\cdots\!57}{22\!\cdots\!15}a^{16}+\frac{90\!\cdots\!74}{22\!\cdots\!15}a^{15}-\frac{10\!\cdots\!58}{22\!\cdots\!15}a^{14}+\frac{55\!\cdots\!07}{22\!\cdots\!15}a^{13}+\frac{11\!\cdots\!29}{22\!\cdots\!15}a^{12}-\frac{17\!\cdots\!78}{22\!\cdots\!15}a^{11}-\frac{32\!\cdots\!83}{22\!\cdots\!15}a^{10}+\frac{20\!\cdots\!83}{22\!\cdots\!15}a^{9}-\frac{16\!\cdots\!84}{44\!\cdots\!83}a^{8}-\frac{10\!\cdots\!07}{22\!\cdots\!15}a^{7}+\frac{17\!\cdots\!23}{44\!\cdots\!83}a^{6}+\frac{64\!\cdots\!54}{22\!\cdots\!15}a^{5}-\frac{34\!\cdots\!31}{22\!\cdots\!15}a^{4}+\frac{96\!\cdots\!03}{22\!\cdots\!15}a^{3}+\frac{86\!\cdots\!66}{47\!\cdots\!45}a^{2}-\frac{50\!\cdots\!29}{22\!\cdots\!15}a-\frac{21\!\cdots\!59}{44\!\cdots\!83}$, $\frac{82\!\cdots\!28}{11\!\cdots\!75}a^{26}-\frac{25\!\cdots\!13}{11\!\cdots\!75}a^{25}+\frac{15\!\cdots\!34}{22\!\cdots\!15}a^{24}+\frac{26\!\cdots\!99}{11\!\cdots\!75}a^{23}-\frac{30\!\cdots\!06}{11\!\cdots\!75}a^{22}+\frac{46\!\cdots\!29}{11\!\cdots\!75}a^{21}+\frac{84\!\cdots\!97}{44\!\cdots\!83}a^{20}+\frac{21\!\cdots\!53}{11\!\cdots\!75}a^{19}+\frac{20\!\cdots\!88}{23\!\cdots\!25}a^{18}-\frac{16\!\cdots\!32}{11\!\cdots\!75}a^{17}-\frac{58\!\cdots\!93}{11\!\cdots\!75}a^{16}+\frac{14\!\cdots\!16}{11\!\cdots\!75}a^{15}+\frac{90\!\cdots\!18}{11\!\cdots\!75}a^{14}-\frac{35\!\cdots\!62}{11\!\cdots\!75}a^{13}+\frac{17\!\cdots\!27}{11\!\cdots\!75}a^{12}+\frac{35\!\cdots\!48}{11\!\cdots\!75}a^{11}-\frac{30\!\cdots\!87}{11\!\cdots\!75}a^{10}-\frac{27\!\cdots\!43}{11\!\cdots\!75}a^{9}+\frac{66\!\cdots\!18}{22\!\cdots\!15}a^{8}+\frac{17\!\cdots\!39}{11\!\cdots\!75}a^{7}-\frac{92\!\cdots\!96}{11\!\cdots\!75}a^{6}-\frac{21\!\cdots\!51}{11\!\cdots\!75}a^{5}-\frac{14\!\cdots\!03}{11\!\cdots\!75}a^{4}+\frac{88\!\cdots\!64}{11\!\cdots\!75}a^{3}-\frac{14\!\cdots\!12}{23\!\cdots\!25}a^{2}-\frac{18\!\cdots\!44}{11\!\cdots\!75}a+\frac{44\!\cdots\!39}{11\!\cdots\!75}$, $\frac{72\!\cdots\!52}{11\!\cdots\!75}a^{26}+\frac{22\!\cdots\!73}{11\!\cdots\!75}a^{25}+\frac{12\!\cdots\!23}{22\!\cdots\!15}a^{24}-\frac{12\!\cdots\!99}{11\!\cdots\!75}a^{23}+\frac{73\!\cdots\!21}{11\!\cdots\!75}a^{22}+\frac{52\!\cdots\!81}{11\!\cdots\!75}a^{21}-\frac{89\!\cdots\!44}{22\!\cdots\!15}a^{20}+\frac{49\!\cdots\!22}{11\!\cdots\!75}a^{19}-\frac{11\!\cdots\!18}{23\!\cdots\!25}a^{18}-\frac{23\!\cdots\!38}{11\!\cdots\!75}a^{17}+\frac{50\!\cdots\!63}{11\!\cdots\!75}a^{16}-\frac{58\!\cdots\!76}{11\!\cdots\!75}a^{15}-\frac{10\!\cdots\!88}{11\!\cdots\!75}a^{14}+\frac{95\!\cdots\!77}{11\!\cdots\!75}a^{13}+\frac{16\!\cdots\!88}{11\!\cdots\!75}a^{12}-\frac{18\!\cdots\!28}{11\!\cdots\!75}a^{11}+\frac{17\!\cdots\!82}{11\!\cdots\!75}a^{10}+\frac{71\!\cdots\!93}{11\!\cdots\!75}a^{9}-\frac{39\!\cdots\!92}{22\!\cdots\!15}a^{8}+\frac{19\!\cdots\!76}{11\!\cdots\!75}a^{7}+\frac{38\!\cdots\!41}{11\!\cdots\!75}a^{6}-\frac{20\!\cdots\!19}{11\!\cdots\!75}a^{5}+\frac{64\!\cdots\!88}{11\!\cdots\!75}a^{4}+\frac{58\!\cdots\!66}{11\!\cdots\!75}a^{3}-\frac{53\!\cdots\!18}{23\!\cdots\!25}a^{2}+\frac{56\!\cdots\!74}{11\!\cdots\!75}a+\frac{28\!\cdots\!91}{11\!\cdots\!75}$, $\frac{65\!\cdots\!49}{11\!\cdots\!75}a^{26}-\frac{12\!\cdots\!71}{11\!\cdots\!75}a^{25}+\frac{63\!\cdots\!83}{22\!\cdots\!15}a^{24}+\frac{80\!\cdots\!23}{11\!\cdots\!75}a^{23}-\frac{20\!\cdots\!57}{11\!\cdots\!75}a^{22}+\frac{16\!\cdots\!13}{11\!\cdots\!75}a^{21}+\frac{15\!\cdots\!89}{22\!\cdots\!15}a^{20}-\frac{82\!\cdots\!59}{11\!\cdots\!75}a^{19}+\frac{23\!\cdots\!36}{23\!\cdots\!25}a^{18}-\frac{50\!\cdots\!24}{11\!\cdots\!75}a^{17}-\frac{19\!\cdots\!86}{11\!\cdots\!75}a^{16}+\frac{73\!\cdots\!17}{11\!\cdots\!75}a^{15}-\frac{24\!\cdots\!24}{11\!\cdots\!75}a^{14}-\frac{14\!\cdots\!29}{11\!\cdots\!75}a^{13}+\frac{18\!\cdots\!49}{11\!\cdots\!75}a^{12}+\frac{41\!\cdots\!36}{11\!\cdots\!75}a^{11}-\frac{30\!\cdots\!99}{11\!\cdots\!75}a^{10}+\frac{15\!\cdots\!94}{11\!\cdots\!75}a^{9}+\frac{35\!\cdots\!46}{22\!\cdots\!15}a^{8}-\frac{26\!\cdots\!07}{11\!\cdots\!75}a^{7}+\frac{39\!\cdots\!43}{11\!\cdots\!75}a^{6}+\frac{10\!\cdots\!18}{11\!\cdots\!75}a^{5}-\frac{67\!\cdots\!56}{11\!\cdots\!75}a^{4}-\frac{23\!\cdots\!97}{11\!\cdots\!75}a^{3}+\frac{54\!\cdots\!91}{23\!\cdots\!25}a^{2}-\frac{83\!\cdots\!68}{11\!\cdots\!75}a-\frac{96\!\cdots\!07}{11\!\cdots\!75}$, $\frac{22\!\cdots\!46}{11\!\cdots\!75}a^{26}+\frac{54\!\cdots\!29}{11\!\cdots\!75}a^{25}+\frac{41\!\cdots\!64}{22\!\cdots\!15}a^{24}-\frac{17\!\cdots\!47}{11\!\cdots\!75}a^{23}+\frac{12\!\cdots\!38}{11\!\cdots\!75}a^{22}+\frac{13\!\cdots\!58}{11\!\cdots\!75}a^{21}+\frac{88\!\cdots\!44}{22\!\cdots\!15}a^{20}+\frac{22\!\cdots\!16}{11\!\cdots\!75}a^{19}-\frac{44\!\cdots\!54}{23\!\cdots\!25}a^{18}-\frac{43\!\cdots\!89}{11\!\cdots\!75}a^{17}+\frac{96\!\cdots\!14}{11\!\cdots\!75}a^{16}+\frac{55\!\cdots\!12}{11\!\cdots\!75}a^{15}-\frac{16\!\cdots\!59}{11\!\cdots\!75}a^{14}-\frac{28\!\cdots\!59}{11\!\cdots\!75}a^{13}+\frac{53\!\cdots\!09}{11\!\cdots\!75}a^{12}-\frac{27\!\cdots\!44}{11\!\cdots\!75}a^{11}-\frac{56\!\cdots\!19}{11\!\cdots\!75}a^{10}-\frac{19\!\cdots\!66}{11\!\cdots\!75}a^{9}+\frac{64\!\cdots\!63}{44\!\cdots\!83}a^{8}+\frac{36\!\cdots\!03}{11\!\cdots\!75}a^{7}-\frac{34\!\cdots\!97}{11\!\cdots\!75}a^{6}-\frac{20\!\cdots\!27}{11\!\cdots\!75}a^{5}+\frac{11\!\cdots\!34}{11\!\cdots\!75}a^{4}-\frac{10\!\cdots\!27}{11\!\cdots\!75}a^{3}-\frac{14\!\cdots\!79}{23\!\cdots\!25}a^{2}+\frac{23\!\cdots\!87}{11\!\cdots\!75}a+\frac{26\!\cdots\!33}{11\!\cdots\!75}$
| 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: | \( 5251358132.129515 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 5251358132.129515 \cdot 1}{2\cdot\sqrt{4852102490441335701504000000000000000000}}\cr\approx \mathstrut & 1.14162919370858 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 9*x^25 + 17*x^24 - 6*x^23 - 86*x^22 + 42*x^21 - 26*x^20 + 2*x^19 + 222*x^18 - 620*x^17 - 136*x^16 + 1722*x^15 - 1022*x^14 - 1740*x^13 + 3140*x^12 + 278*x^11 - 3510*x^10 + 1746*x^9 + 1702*x^8 - 1886*x^7 - 6*x^6 + 1068*x^5 - 272*x^4 - 230*x^3 + 166*x^2 + 10*x - 58)
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^27 - 3*x^26 - 9*x^25 + 17*x^24 - 6*x^23 - 86*x^22 + 42*x^21 - 26*x^20 + 2*x^19 + 222*x^18 - 620*x^17 - 136*x^16 + 1722*x^15 - 1022*x^14 - 1740*x^13 + 3140*x^12 + 278*x^11 - 3510*x^10 + 1746*x^9 + 1702*x^8 - 1886*x^7 - 6*x^6 + 1068*x^5 - 272*x^4 - 230*x^3 + 166*x^2 + 10*x - 58, 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^27 - 3*x^26 - 9*x^25 + 17*x^24 - 6*x^23 - 86*x^22 + 42*x^21 - 26*x^20 + 2*x^19 + 222*x^18 - 620*x^17 - 136*x^16 + 1722*x^15 - 1022*x^14 - 1740*x^13 + 3140*x^12 + 278*x^11 - 3510*x^10 + 1746*x^9 + 1702*x^8 - 1886*x^7 - 6*x^6 + 1068*x^5 - 272*x^4 - 230*x^3 + 166*x^2 + 10*x - 58);
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^27 - 3*x^26 - 9*x^25 + 17*x^24 - 6*x^23 - 86*x^22 + 42*x^21 - 26*x^20 + 2*x^19 + 222*x^18 - 620*x^17 - 136*x^16 + 1722*x^15 - 1022*x^14 - 1740*x^13 + 3140*x^12 + 278*x^11 - 3510*x^10 + 1746*x^9 + 1702*x^8 - 1886*x^7 - 6*x^6 + 1068*x^5 - 272*x^4 - 230*x^3 + 166*x^2 + 10*x - 58);
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
$\He_3:\GL(2,3)$ (as 27T294):
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 1296 |
| The 18 conjugacy class representatives for $\He_3:\GL(2,3)$ |
| Character table for $\He_3:\GL(2,3)$ |
Intermediate fields
| 9.3.2239488000000.1 |
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
| Degree 27 siblings: | data not computed |
| Degree 36 siblings: | 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 | R | R | R | ${\href{/padicField/7.9.0.1}{9} }^{3}$ | ${\href{/padicField/11.8.0.1}{8} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{4}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.8.0.1}{8} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.9.0.1}{9} }^{3}$ | ${\href{/padicField/41.8.0.1}{8} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.8.0.1}{8} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| Deg $24$ | $24$ | $1$ | $50$ | ||||
|
\(3\)
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| Deg $18$ | $6$ | $3$ | $21$ | ||||
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.8.6.4 | $x^{8} - 20 x^{4} + 50$ | $4$ | $2$ | $6$ | $C_8$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.4 | $x^{8} - 20 x^{4} + 50$ | $4$ | $2$ | $6$ | $C_8$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.4 | $x^{8} - 20 x^{4} + 50$ | $4$ | $2$ | $6$ | $C_8$ | $[\ ]_{4}^{2}$ |