Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 8*x^27 + 39*x^26 - 147*x^25 + 484*x^24 - 1461*x^23 + 4001*x^22 - 10052*x^21 + 23841*x^20 - 52926*x^19 + 109173*x^18 - 208723*x^17 + 369896*x^16 - 608215*x^15 + 913744*x^14 - 1246890*x^13 + 1584122*x^12 - 1829692*x^11 + 1878311*x^10 - 1777347*x^9 + 1518093*x^8 - 1036936*x^7 + 598732*x^6 - 344364*x^5 + 177291*x^4 - 79332*x^3 + 46992*x^2 - 15367*x + 6413)
gp: K = bnfinit(y^28 - 8*y^27 + 39*y^26 - 147*y^25 + 484*y^24 - 1461*y^23 + 4001*y^22 - 10052*y^21 + 23841*y^20 - 52926*y^19 + 109173*y^18 - 208723*y^17 + 369896*y^16 - 608215*y^15 + 913744*y^14 - 1246890*y^13 + 1584122*y^12 - 1829692*y^11 + 1878311*y^10 - 1777347*y^9 + 1518093*y^8 - 1036936*y^7 + 598732*y^6 - 344364*y^5 + 177291*y^4 - 79332*y^3 + 46992*y^2 - 15367*y + 6413, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 8*x^27 + 39*x^26 - 147*x^25 + 484*x^24 - 1461*x^23 + 4001*x^22 - 10052*x^21 + 23841*x^20 - 52926*x^19 + 109173*x^18 - 208723*x^17 + 369896*x^16 - 608215*x^15 + 913744*x^14 - 1246890*x^13 + 1584122*x^12 - 1829692*x^11 + 1878311*x^10 - 1777347*x^9 + 1518093*x^8 - 1036936*x^7 + 598732*x^6 - 344364*x^5 + 177291*x^4 - 79332*x^3 + 46992*x^2 - 15367*x + 6413);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 8*x^27 + 39*x^26 - 147*x^25 + 484*x^24 - 1461*x^23 + 4001*x^22 - 10052*x^21 + 23841*x^20 - 52926*x^19 + 109173*x^18 - 208723*x^17 + 369896*x^16 - 608215*x^15 + 913744*x^14 - 1246890*x^13 + 1584122*x^12 - 1829692*x^11 + 1878311*x^10 - 1777347*x^9 + 1518093*x^8 - 1036936*x^7 + 598732*x^6 - 344364*x^5 + 177291*x^4 - 79332*x^3 + 46992*x^2 - 15367*x + 6413)
\( x^{28} - 8 x^{27} + 39 x^{26} - 147 x^{25} + 484 x^{24} - 1461 x^{23} + 4001 x^{22} - 10052 x^{21} + \cdots + 6413 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(84980457635217037392318371738994676821454381\)
\(\medspace = 11^{14}\cdot 181^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(37.06\) | 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: | $11^{1/2}181^{1/2}\approx 44.62062303464621$ | ||
| Ramified primes: |
\(11\), \(181\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{181}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11}a^{16}+\frac{2}{11}a^{15}+\frac{2}{11}a^{14}+\frac{2}{11}a^{13}-\frac{2}{11}a^{11}+\frac{1}{11}a^{10}+\frac{4}{11}a^{9}-\frac{3}{11}a^{8}+\frac{1}{11}a^{7}-\frac{2}{11}a^{6}-\frac{5}{11}a^{5}-\frac{4}{11}a^{4}-\frac{3}{11}a^{3}+\frac{4}{11}a^{2}$, $\frac{1}{11}a^{17}-\frac{2}{11}a^{15}-\frac{2}{11}a^{14}-\frac{4}{11}a^{13}-\frac{2}{11}a^{12}+\frac{5}{11}a^{11}+\frac{2}{11}a^{10}-\frac{4}{11}a^{8}-\frac{4}{11}a^{7}-\frac{1}{11}a^{6}-\frac{5}{11}a^{5}+\frac{5}{11}a^{4}-\frac{1}{11}a^{3}+\frac{3}{11}a^{2}$, $\frac{1}{11}a^{18}+\frac{2}{11}a^{15}+\frac{2}{11}a^{13}+\frac{5}{11}a^{12}-\frac{2}{11}a^{11}+\frac{2}{11}a^{10}+\frac{4}{11}a^{9}+\frac{1}{11}a^{8}+\frac{1}{11}a^{7}+\frac{2}{11}a^{6}-\frac{5}{11}a^{5}+\frac{2}{11}a^{4}-\frac{3}{11}a^{3}-\frac{3}{11}a^{2}$, $\frac{1}{11}a^{19}-\frac{4}{11}a^{15}-\frac{2}{11}a^{14}+\frac{1}{11}a^{13}-\frac{2}{11}a^{12}-\frac{5}{11}a^{11}+\frac{2}{11}a^{10}+\frac{4}{11}a^{9}-\frac{4}{11}a^{8}-\frac{1}{11}a^{6}+\frac{1}{11}a^{5}+\frac{5}{11}a^{4}+\frac{3}{11}a^{3}+\frac{3}{11}a^{2}$, $\frac{1}{11}a^{20}-\frac{5}{11}a^{15}-\frac{2}{11}a^{14}-\frac{5}{11}a^{13}-\frac{5}{11}a^{12}+\frac{5}{11}a^{11}-\frac{3}{11}a^{10}+\frac{1}{11}a^{9}-\frac{1}{11}a^{8}+\frac{3}{11}a^{7}+\frac{4}{11}a^{6}-\frac{4}{11}a^{5}-\frac{2}{11}a^{4}+\frac{2}{11}a^{3}+\frac{5}{11}a^{2}$, $\frac{1}{11}a^{21}-\frac{3}{11}a^{15}+\frac{5}{11}a^{14}+\frac{5}{11}a^{13}+\frac{5}{11}a^{12}-\frac{2}{11}a^{11}-\frac{5}{11}a^{10}-\frac{3}{11}a^{9}-\frac{1}{11}a^{8}-\frac{2}{11}a^{7}-\frac{3}{11}a^{6}-\frac{5}{11}a^{5}+\frac{4}{11}a^{4}+\frac{1}{11}a^{3}-\frac{2}{11}a^{2}$, $\frac{1}{121}a^{22}+\frac{2}{121}a^{21}-\frac{4}{121}a^{20}+\frac{1}{121}a^{19}-\frac{3}{121}a^{18}+\frac{4}{121}a^{17}-\frac{3}{121}a^{16}+\frac{23}{121}a^{15}-\frac{42}{121}a^{14}+\frac{14}{121}a^{13}+\frac{3}{121}a^{12}+\frac{36}{121}a^{11}+\frac{14}{121}a^{10}-\frac{8}{121}a^{9}+\frac{10}{121}a^{8}+\frac{6}{121}a^{7}+\frac{50}{121}a^{6}+\frac{6}{121}a^{5}+\frac{58}{121}a^{4}-\frac{5}{11}a^{3}-\frac{2}{11}a^{2}$, $\frac{1}{121}a^{23}+\frac{3}{121}a^{21}-\frac{2}{121}a^{20}-\frac{5}{121}a^{19}-\frac{1}{121}a^{18}-\frac{4}{121}a^{16}-\frac{5}{11}a^{15}-\frac{34}{121}a^{14}-\frac{47}{121}a^{13}-\frac{58}{121}a^{12}+\frac{8}{121}a^{11}+\frac{30}{121}a^{10}+\frac{48}{121}a^{9}+\frac{30}{121}a^{8}+\frac{16}{121}a^{7}-\frac{17}{121}a^{6}-\frac{42}{121}a^{5}+\frac{60}{121}a^{4}-\frac{4}{11}a^{3}+\frac{2}{11}a^{2}$, $\frac{1}{121}a^{24}+\frac{3}{121}a^{21}-\frac{4}{121}a^{20}-\frac{4}{121}a^{19}-\frac{2}{121}a^{18}-\frac{5}{121}a^{17}-\frac{2}{121}a^{16}-\frac{37}{121}a^{15}-\frac{20}{121}a^{14}+\frac{32}{121}a^{13}+\frac{32}{121}a^{12}-\frac{45}{121}a^{11}+\frac{28}{121}a^{10}+\frac{21}{121}a^{9}+\frac{41}{121}a^{8}+\frac{20}{121}a^{7}-\frac{27}{121}a^{6}+\frac{53}{121}a^{5}-\frac{53}{121}a^{4}-\frac{5}{11}a^{3}-\frac{1}{11}a^{2}$, $\frac{1}{1573}a^{25}-\frac{6}{1573}a^{23}+\frac{6}{1573}a^{22}+\frac{61}{1573}a^{21}-\frac{70}{1573}a^{20}-\frac{35}{1573}a^{19}-\frac{30}{1573}a^{18}+\frac{43}{1573}a^{17}-\frac{6}{143}a^{16}+\frac{60}{1573}a^{15}+\frac{2}{13}a^{14}-\frac{711}{1573}a^{13}+\frac{620}{1573}a^{12}+\frac{43}{143}a^{11}-\frac{700}{1573}a^{10}-\frac{733}{1573}a^{9}+\frac{101}{1573}a^{8}-\frac{171}{1573}a^{7}+\frac{492}{1573}a^{6}+\frac{74}{1573}a^{5}-\frac{439}{1573}a^{4}-\frac{3}{13}a^{3}+\frac{40}{143}a^{2}-\frac{5}{13}a+\frac{2}{13}$, $\frac{1}{72032389}a^{26}-\frac{7643}{72032389}a^{25}+\frac{189586}{72032389}a^{24}+\frac{985}{503723}a^{23}+\frac{11195}{72032389}a^{22}-\frac{2706050}{72032389}a^{21}-\frac{2374503}{72032389}a^{20}+\frac{85144}{5540953}a^{19}-\frac{222363}{5540953}a^{18}-\frac{2385}{284713}a^{17}-\frac{310940}{72032389}a^{16}-\frac{21195444}{72032389}a^{15}+\frac{14028182}{72032389}a^{14}+\frac{397821}{5540953}a^{13}+\frac{9147971}{72032389}a^{12}+\frac{9858193}{72032389}a^{11}-\frac{29140413}{72032389}a^{10}-\frac{17455628}{72032389}a^{9}+\frac{32715938}{72032389}a^{8}-\frac{14910042}{72032389}a^{7}-\frac{31894024}{72032389}a^{6}-\frac{30330158}{72032389}a^{5}+\frac{26940996}{72032389}a^{4}-\frac{2411116}{6548399}a^{3}-\frac{3263288}{6548399}a^{2}-\frac{2434}{595309}a-\frac{280057}{595309}$, $\frac{1}{69\!\cdots\!61}a^{27}-\frac{24\!\cdots\!18}{69\!\cdots\!61}a^{26}+\frac{10\!\cdots\!71}{69\!\cdots\!61}a^{25}+\frac{18\!\cdots\!33}{69\!\cdots\!61}a^{24}-\frac{15\!\cdots\!69}{69\!\cdots\!61}a^{23}+\frac{45\!\cdots\!47}{13\!\cdots\!37}a^{22}-\frac{28\!\cdots\!64}{69\!\cdots\!61}a^{21}-\frac{19\!\cdots\!64}{69\!\cdots\!61}a^{20}-\frac{59\!\cdots\!51}{69\!\cdots\!61}a^{19}+\frac{14\!\cdots\!69}{69\!\cdots\!61}a^{18}+\frac{11\!\cdots\!83}{69\!\cdots\!61}a^{17}+\frac{15\!\cdots\!45}{69\!\cdots\!61}a^{16}+\frac{80\!\cdots\!75}{27\!\cdots\!37}a^{15}+\frac{29\!\cdots\!99}{69\!\cdots\!61}a^{14}+\frac{33\!\cdots\!48}{69\!\cdots\!61}a^{13}+\frac{32\!\cdots\!20}{69\!\cdots\!61}a^{12}+\frac{34\!\cdots\!83}{69\!\cdots\!61}a^{11}+\frac{17\!\cdots\!94}{63\!\cdots\!51}a^{10}+\frac{33\!\cdots\!60}{69\!\cdots\!61}a^{9}+\frac{10\!\cdots\!84}{69\!\cdots\!61}a^{8}-\frac{11\!\cdots\!04}{30\!\cdots\!07}a^{7}+\frac{11\!\cdots\!07}{69\!\cdots\!61}a^{6}+\frac{98\!\cdots\!23}{69\!\cdots\!61}a^{5}-\frac{25\!\cdots\!30}{69\!\cdots\!61}a^{4}+\frac{19\!\cdots\!14}{61\!\cdots\!17}a^{3}-\frac{45\!\cdots\!22}{27\!\cdots\!37}a^{2}-\frac{12\!\cdots\!21}{57\!\cdots\!41}a-\frac{25\!\cdots\!08}{10\!\cdots\!97}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $11$ |
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: | $13$ | 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: |
$\frac{27\!\cdots\!12}{69\!\cdots\!61}a^{27}-\frac{24\!\cdots\!81}{69\!\cdots\!61}a^{26}+\frac{12\!\cdots\!11}{69\!\cdots\!61}a^{25}-\frac{39\!\cdots\!15}{53\!\cdots\!97}a^{24}+\frac{17\!\cdots\!35}{69\!\cdots\!61}a^{23}-\frac{98\!\cdots\!41}{13\!\cdots\!37}a^{22}+\frac{14\!\cdots\!91}{69\!\cdots\!61}a^{21}-\frac{37\!\cdots\!60}{69\!\cdots\!61}a^{20}+\frac{89\!\cdots\!34}{69\!\cdots\!61}a^{19}-\frac{20\!\cdots\!08}{69\!\cdots\!61}a^{18}+\frac{42\!\cdots\!84}{69\!\cdots\!61}a^{17}-\frac{35\!\cdots\!09}{30\!\cdots\!07}a^{16}+\frac{13\!\cdots\!74}{63\!\cdots\!51}a^{15}-\frac{24\!\cdots\!18}{69\!\cdots\!61}a^{14}+\frac{37\!\cdots\!02}{69\!\cdots\!61}a^{13}-\frac{52\!\cdots\!01}{69\!\cdots\!61}a^{12}+\frac{66\!\cdots\!71}{69\!\cdots\!61}a^{11}-\frac{71\!\cdots\!63}{63\!\cdots\!51}a^{10}+\frac{81\!\cdots\!51}{69\!\cdots\!61}a^{9}-\frac{75\!\cdots\!74}{69\!\cdots\!61}a^{8}+\frac{64\!\cdots\!14}{69\!\cdots\!61}a^{7}-\frac{45\!\cdots\!08}{69\!\cdots\!61}a^{6}+\frac{17\!\cdots\!52}{53\!\cdots\!97}a^{5}-\frac{89\!\cdots\!58}{69\!\cdots\!61}a^{4}+\frac{44\!\cdots\!20}{61\!\cdots\!17}a^{3}-\frac{15\!\cdots\!66}{63\!\cdots\!51}a^{2}+\frac{44\!\cdots\!81}{57\!\cdots\!41}a+\frac{75\!\cdots\!55}{10\!\cdots\!97}$, $\frac{10\!\cdots\!49}{69\!\cdots\!61}a^{27}-\frac{75\!\cdots\!07}{69\!\cdots\!61}a^{26}+\frac{26\!\cdots\!37}{53\!\cdots\!97}a^{25}-\frac{12\!\cdots\!00}{69\!\cdots\!61}a^{24}+\frac{31\!\cdots\!08}{53\!\cdots\!97}a^{23}-\frac{17\!\cdots\!63}{10\!\cdots\!49}a^{22}+\frac{32\!\cdots\!70}{69\!\cdots\!61}a^{21}-\frac{34\!\cdots\!11}{30\!\cdots\!07}a^{20}+\frac{18\!\cdots\!28}{69\!\cdots\!61}a^{19}-\frac{40\!\cdots\!08}{69\!\cdots\!61}a^{18}+\frac{80\!\cdots\!07}{69\!\cdots\!61}a^{17}-\frac{15\!\cdots\!09}{69\!\cdots\!61}a^{16}+\frac{23\!\cdots\!29}{63\!\cdots\!51}a^{15}-\frac{41\!\cdots\!77}{69\!\cdots\!61}a^{14}+\frac{59\!\cdots\!72}{69\!\cdots\!61}a^{13}-\frac{78\!\cdots\!33}{69\!\cdots\!61}a^{12}+\frac{96\!\cdots\!38}{69\!\cdots\!61}a^{11}-\frac{95\!\cdots\!05}{63\!\cdots\!51}a^{10}+\frac{77\!\cdots\!17}{53\!\cdots\!97}a^{9}-\frac{91\!\cdots\!08}{69\!\cdots\!61}a^{8}+\frac{71\!\cdots\!04}{69\!\cdots\!61}a^{7}-\frac{37\!\cdots\!01}{69\!\cdots\!61}a^{6}+\frac{20\!\cdots\!57}{69\!\cdots\!61}a^{5}-\frac{13\!\cdots\!74}{69\!\cdots\!61}a^{4}+\frac{34\!\cdots\!82}{61\!\cdots\!17}a^{3}-\frac{26\!\cdots\!01}{63\!\cdots\!51}a^{2}+\frac{13\!\cdots\!27}{57\!\cdots\!41}a-\frac{78\!\cdots\!48}{10\!\cdots\!97}$, $\frac{12\!\cdots\!50}{69\!\cdots\!61}a^{27}+\frac{33\!\cdots\!35}{69\!\cdots\!61}a^{26}-\frac{30\!\cdots\!75}{69\!\cdots\!61}a^{25}+\frac{15\!\cdots\!73}{69\!\cdots\!61}a^{24}-\frac{57\!\cdots\!15}{69\!\cdots\!61}a^{23}+\frac{35\!\cdots\!38}{13\!\cdots\!37}a^{22}-\frac{57\!\cdots\!55}{69\!\cdots\!61}a^{21}+\frac{15\!\cdots\!21}{69\!\cdots\!61}a^{20}-\frac{39\!\cdots\!53}{69\!\cdots\!61}a^{19}+\frac{92\!\cdots\!47}{69\!\cdots\!61}a^{18}-\frac{20\!\cdots\!24}{69\!\cdots\!61}a^{17}+\frac{41\!\cdots\!15}{69\!\cdots\!61}a^{16}-\frac{71\!\cdots\!74}{63\!\cdots\!51}a^{15}+\frac{13\!\cdots\!33}{69\!\cdots\!61}a^{14}-\frac{22\!\cdots\!85}{69\!\cdots\!61}a^{13}+\frac{32\!\cdots\!42}{69\!\cdots\!61}a^{12}-\frac{43\!\cdots\!58}{69\!\cdots\!61}a^{11}+\frac{43\!\cdots\!54}{57\!\cdots\!41}a^{10}-\frac{58\!\cdots\!52}{69\!\cdots\!61}a^{9}+\frac{55\!\cdots\!00}{69\!\cdots\!61}a^{8}-\frac{48\!\cdots\!91}{69\!\cdots\!61}a^{7}+\frac{28\!\cdots\!36}{53\!\cdots\!97}a^{6}-\frac{19\!\cdots\!33}{69\!\cdots\!61}a^{5}+\frac{52\!\cdots\!69}{69\!\cdots\!61}a^{4}-\frac{18\!\cdots\!94}{61\!\cdots\!17}a^{3}+\frac{16\!\cdots\!33}{63\!\cdots\!51}a^{2}-\frac{10\!\cdots\!75}{57\!\cdots\!41}a+\frac{69\!\cdots\!51}{10\!\cdots\!97}$, $\frac{47\!\cdots\!60}{69\!\cdots\!61}a^{27}-\frac{43\!\cdots\!68}{69\!\cdots\!61}a^{26}+\frac{22\!\cdots\!35}{69\!\cdots\!61}a^{25}-\frac{91\!\cdots\!15}{69\!\cdots\!61}a^{24}+\frac{30\!\cdots\!23}{69\!\cdots\!61}a^{23}-\frac{17\!\cdots\!84}{13\!\cdots\!37}a^{22}+\frac{20\!\cdots\!93}{53\!\cdots\!97}a^{21}-\frac{68\!\cdots\!48}{69\!\cdots\!61}a^{20}+\frac{16\!\cdots\!62}{69\!\cdots\!61}a^{19}-\frac{37\!\cdots\!21}{69\!\cdots\!61}a^{18}+\frac{78\!\cdots\!76}{69\!\cdots\!61}a^{17}-\frac{15\!\cdots\!14}{69\!\cdots\!61}a^{16}+\frac{25\!\cdots\!71}{63\!\cdots\!51}a^{15}-\frac{46\!\cdots\!97}{69\!\cdots\!61}a^{14}+\frac{71\!\cdots\!85}{69\!\cdots\!61}a^{13}-\frac{10\!\cdots\!90}{69\!\cdots\!61}a^{12}+\frac{12\!\cdots\!15}{69\!\cdots\!61}a^{11}-\frac{13\!\cdots\!48}{63\!\cdots\!51}a^{10}+\frac{70\!\cdots\!31}{30\!\cdots\!07}a^{9}-\frac{15\!\cdots\!97}{69\!\cdots\!61}a^{8}+\frac{13\!\cdots\!48}{69\!\cdots\!61}a^{7}-\frac{96\!\cdots\!45}{69\!\cdots\!61}a^{6}+\frac{53\!\cdots\!74}{69\!\cdots\!61}a^{5}-\frac{25\!\cdots\!38}{69\!\cdots\!61}a^{4}+\frac{10\!\cdots\!20}{61\!\cdots\!17}a^{3}-\frac{41\!\cdots\!48}{63\!\cdots\!51}a^{2}+\frac{16\!\cdots\!29}{57\!\cdots\!41}a-\frac{26\!\cdots\!25}{10\!\cdots\!97}$, $\frac{14\!\cdots\!28}{69\!\cdots\!61}a^{27}-\frac{10\!\cdots\!41}{69\!\cdots\!61}a^{26}+\frac{40\!\cdots\!33}{53\!\cdots\!97}a^{25}-\frac{14\!\cdots\!36}{53\!\cdots\!97}a^{24}+\frac{62\!\cdots\!46}{69\!\cdots\!61}a^{23}-\frac{35\!\cdots\!19}{13\!\cdots\!37}a^{22}+\frac{50\!\cdots\!50}{69\!\cdots\!61}a^{21}-\frac{12\!\cdots\!10}{69\!\cdots\!61}a^{20}+\frac{29\!\cdots\!10}{69\!\cdots\!61}a^{19}-\frac{65\!\cdots\!05}{69\!\cdots\!61}a^{18}+\frac{13\!\cdots\!22}{69\!\cdots\!61}a^{17}-\frac{24\!\cdots\!60}{69\!\cdots\!61}a^{16}+\frac{39\!\cdots\!90}{63\!\cdots\!51}a^{15}-\frac{70\!\cdots\!69}{69\!\cdots\!61}a^{14}+\frac{10\!\cdots\!59}{69\!\cdots\!61}a^{13}-\frac{13\!\cdots\!21}{69\!\cdots\!61}a^{12}+\frac{16\!\cdots\!93}{69\!\cdots\!61}a^{11}-\frac{16\!\cdots\!47}{63\!\cdots\!51}a^{10}+\frac{18\!\cdots\!68}{69\!\cdots\!61}a^{9}-\frac{16\!\cdots\!47}{69\!\cdots\!61}a^{8}+\frac{12\!\cdots\!77}{69\!\cdots\!61}a^{7}-\frac{56\!\cdots\!73}{53\!\cdots\!97}a^{6}+\frac{31\!\cdots\!16}{69\!\cdots\!61}a^{5}-\frac{17\!\cdots\!28}{69\!\cdots\!61}a^{4}+\frac{10\!\cdots\!47}{61\!\cdots\!17}a^{3}-\frac{52\!\cdots\!94}{63\!\cdots\!51}a^{2}+\frac{18\!\cdots\!77}{57\!\cdots\!41}a-\frac{58\!\cdots\!24}{10\!\cdots\!97}$, $\frac{16\!\cdots\!29}{69\!\cdots\!61}a^{27}-\frac{79\!\cdots\!42}{69\!\cdots\!61}a^{26}+\frac{27\!\cdots\!06}{69\!\cdots\!61}a^{25}-\frac{76\!\cdots\!95}{69\!\cdots\!61}a^{24}+\frac{21\!\cdots\!95}{69\!\cdots\!61}a^{23}-\frac{10\!\cdots\!40}{13\!\cdots\!37}a^{22}+\frac{91\!\cdots\!53}{53\!\cdots\!97}a^{21}-\frac{24\!\cdots\!79}{69\!\cdots\!61}a^{20}+\frac{39\!\cdots\!90}{53\!\cdots\!97}a^{19}-\frac{85\!\cdots\!23}{69\!\cdots\!61}a^{18}+\frac{12\!\cdots\!05}{69\!\cdots\!61}a^{17}-\frac{13\!\cdots\!38}{69\!\cdots\!61}a^{16}+\frac{11\!\cdots\!34}{63\!\cdots\!51}a^{15}-\frac{77\!\cdots\!42}{69\!\cdots\!61}a^{14}-\frac{20\!\cdots\!83}{69\!\cdots\!61}a^{13}+\frac{18\!\cdots\!69}{69\!\cdots\!61}a^{12}+\frac{11\!\cdots\!34}{69\!\cdots\!61}a^{11}-\frac{12\!\cdots\!81}{57\!\cdots\!41}a^{10}+\frac{37\!\cdots\!07}{69\!\cdots\!61}a^{9}-\frac{89\!\cdots\!80}{69\!\cdots\!61}a^{8}+\frac{98\!\cdots\!58}{69\!\cdots\!61}a^{7}-\frac{11\!\cdots\!09}{69\!\cdots\!61}a^{6}+\frac{16\!\cdots\!69}{69\!\cdots\!61}a^{5}-\frac{92\!\cdots\!82}{53\!\cdots\!97}a^{4}+\frac{53\!\cdots\!72}{61\!\cdots\!17}a^{3}-\frac{29\!\cdots\!13}{63\!\cdots\!51}a^{2}+\frac{54\!\cdots\!90}{57\!\cdots\!41}a+\frac{23\!\cdots\!36}{10\!\cdots\!97}$, $\frac{93\!\cdots\!45}{69\!\cdots\!61}a^{27}-\frac{72\!\cdots\!70}{69\!\cdots\!61}a^{26}+\frac{34\!\cdots\!59}{69\!\cdots\!61}a^{25}-\frac{12\!\cdots\!20}{69\!\cdots\!61}a^{24}+\frac{42\!\cdots\!75}{69\!\cdots\!61}a^{23}-\frac{23\!\cdots\!55}{13\!\cdots\!37}a^{22}+\frac{34\!\cdots\!56}{69\!\cdots\!61}a^{21}-\frac{37\!\cdots\!48}{30\!\cdots\!07}a^{20}+\frac{20\!\cdots\!56}{69\!\cdots\!61}a^{19}-\frac{44\!\cdots\!73}{69\!\cdots\!61}a^{18}+\frac{90\!\cdots\!20}{69\!\cdots\!61}a^{17}-\frac{17\!\cdots\!60}{69\!\cdots\!61}a^{16}+\frac{27\!\cdots\!32}{63\!\cdots\!51}a^{15}-\frac{48\!\cdots\!89}{69\!\cdots\!61}a^{14}+\frac{71\!\cdots\!41}{69\!\cdots\!61}a^{13}-\frac{95\!\cdots\!84}{69\!\cdots\!61}a^{12}+\frac{91\!\cdots\!12}{53\!\cdots\!97}a^{11}-\frac{11\!\cdots\!83}{57\!\cdots\!41}a^{10}+\frac{13\!\cdots\!37}{69\!\cdots\!61}a^{9}-\frac{12\!\cdots\!13}{69\!\cdots\!61}a^{8}+\frac{10\!\cdots\!76}{69\!\cdots\!61}a^{7}-\frac{60\!\cdots\!54}{69\!\cdots\!61}a^{6}+\frac{30\!\cdots\!66}{69\!\cdots\!61}a^{5}-\frac{19\!\cdots\!45}{69\!\cdots\!61}a^{4}+\frac{83\!\cdots\!24}{61\!\cdots\!17}a^{3}-\frac{28\!\cdots\!35}{63\!\cdots\!51}a^{2}+\frac{17\!\cdots\!70}{44\!\cdots\!57}a-\frac{10\!\cdots\!94}{10\!\cdots\!97}$, $\frac{95\!\cdots\!94}{69\!\cdots\!61}a^{27}-\frac{54\!\cdots\!52}{69\!\cdots\!61}a^{26}+\frac{18\!\cdots\!99}{69\!\cdots\!61}a^{25}-\frac{46\!\cdots\!05}{69\!\cdots\!61}a^{24}+\frac{10\!\cdots\!29}{69\!\cdots\!61}a^{23}-\frac{38\!\cdots\!64}{13\!\cdots\!37}a^{22}+\frac{21\!\cdots\!21}{69\!\cdots\!61}a^{21}+\frac{34\!\cdots\!76}{69\!\cdots\!61}a^{20}-\frac{23\!\cdots\!56}{69\!\cdots\!61}a^{19}+\frac{90\!\cdots\!75}{69\!\cdots\!61}a^{18}-\frac{21\!\cdots\!64}{53\!\cdots\!97}a^{17}+\frac{75\!\cdots\!81}{69\!\cdots\!61}a^{16}-\frac{15\!\cdots\!72}{63\!\cdots\!51}a^{15}+\frac{35\!\cdots\!98}{69\!\cdots\!61}a^{14}-\frac{65\!\cdots\!24}{69\!\cdots\!61}a^{13}+\frac{10\!\cdots\!06}{69\!\cdots\!61}a^{12}-\frac{15\!\cdots\!58}{69\!\cdots\!61}a^{11}+\frac{19\!\cdots\!66}{63\!\cdots\!51}a^{10}-\frac{26\!\cdots\!13}{69\!\cdots\!61}a^{9}+\frac{26\!\cdots\!58}{69\!\cdots\!61}a^{8}-\frac{25\!\cdots\!92}{69\!\cdots\!61}a^{7}+\frac{22\!\cdots\!80}{69\!\cdots\!61}a^{6}-\frac{14\!\cdots\!01}{69\!\cdots\!61}a^{5}+\frac{49\!\cdots\!36}{69\!\cdots\!61}a^{4}-\frac{22\!\cdots\!52}{61\!\cdots\!17}a^{3}+\frac{22\!\cdots\!74}{63\!\cdots\!51}a^{2}-\frac{16\!\cdots\!89}{57\!\cdots\!41}a+\frac{48\!\cdots\!15}{10\!\cdots\!97}$, $\frac{42\!\cdots\!27}{69\!\cdots\!61}a^{27}-\frac{29\!\cdots\!62}{69\!\cdots\!61}a^{26}+\frac{95\!\cdots\!53}{53\!\cdots\!97}a^{25}-\frac{40\!\cdots\!53}{69\!\cdots\!61}a^{24}+\frac{11\!\cdots\!35}{69\!\cdots\!61}a^{23}-\frac{60\!\cdots\!42}{13\!\cdots\!37}a^{22}+\frac{77\!\cdots\!86}{69\!\cdots\!61}a^{21}-\frac{12\!\cdots\!83}{53\!\cdots\!97}a^{20}+\frac{33\!\cdots\!47}{69\!\cdots\!61}a^{19}-\frac{61\!\cdots\!84}{69\!\cdots\!61}a^{18}+\frac{91\!\cdots\!38}{69\!\cdots\!61}a^{17}-\frac{96\!\cdots\!12}{69\!\cdots\!61}a^{16}+\frac{11\!\cdots\!40}{63\!\cdots\!51}a^{15}+\frac{21\!\cdots\!69}{53\!\cdots\!97}a^{14}-\frac{98\!\cdots\!20}{69\!\cdots\!61}a^{13}+\frac{23\!\cdots\!73}{69\!\cdots\!61}a^{12}-\frac{42\!\cdots\!16}{69\!\cdots\!61}a^{11}+\frac{61\!\cdots\!36}{63\!\cdots\!51}a^{10}-\frac{42\!\cdots\!11}{30\!\cdots\!07}a^{9}+\frac{11\!\cdots\!05}{69\!\cdots\!61}a^{8}-\frac{13\!\cdots\!19}{69\!\cdots\!61}a^{7}+\frac{13\!\cdots\!73}{69\!\cdots\!61}a^{6}-\frac{11\!\cdots\!84}{69\!\cdots\!61}a^{5}+\frac{74\!\cdots\!52}{69\!\cdots\!61}a^{4}-\frac{35\!\cdots\!78}{61\!\cdots\!17}a^{3}+\frac{17\!\cdots\!77}{63\!\cdots\!51}a^{2}-\frac{31\!\cdots\!97}{44\!\cdots\!57}a+\frac{18\!\cdots\!32}{10\!\cdots\!97}$, $\frac{13\!\cdots\!79}{57\!\cdots\!41}a^{27}-\frac{12\!\cdots\!22}{63\!\cdots\!51}a^{26}+\frac{59\!\cdots\!98}{63\!\cdots\!51}a^{25}-\frac{22\!\cdots\!21}{63\!\cdots\!51}a^{24}+\frac{61\!\cdots\!62}{52\!\cdots\!31}a^{23}-\frac{18\!\cdots\!37}{51\!\cdots\!29}a^{22}+\frac{62\!\cdots\!80}{63\!\cdots\!51}a^{21}-\frac{15\!\cdots\!38}{63\!\cdots\!51}a^{20}+\frac{37\!\cdots\!52}{63\!\cdots\!51}a^{19}-\frac{82\!\cdots\!16}{63\!\cdots\!51}a^{18}+\frac{15\!\cdots\!66}{57\!\cdots\!41}a^{17}-\frac{32\!\cdots\!40}{63\!\cdots\!51}a^{16}+\frac{44\!\cdots\!48}{48\!\cdots\!27}a^{15}-\frac{94\!\cdots\!67}{63\!\cdots\!51}a^{14}+\frac{10\!\cdots\!32}{48\!\cdots\!27}a^{13}-\frac{19\!\cdots\!41}{63\!\cdots\!51}a^{12}+\frac{24\!\cdots\!62}{63\!\cdots\!51}a^{11}-\frac{28\!\cdots\!83}{63\!\cdots\!51}a^{10}+\frac{28\!\cdots\!42}{63\!\cdots\!51}a^{9}-\frac{26\!\cdots\!14}{63\!\cdots\!51}a^{8}+\frac{22\!\cdots\!41}{63\!\cdots\!51}a^{7}-\frac{11\!\cdots\!50}{48\!\cdots\!27}a^{6}+\frac{75\!\cdots\!70}{63\!\cdots\!51}a^{5}-\frac{42\!\cdots\!75}{63\!\cdots\!51}a^{4}+\frac{21\!\cdots\!17}{55\!\cdots\!47}a^{3}-\frac{81\!\cdots\!74}{57\!\cdots\!41}a^{2}+\frac{47\!\cdots\!07}{52\!\cdots\!31}a-\frac{46\!\cdots\!89}{98\!\cdots\!27}$, $\frac{11\!\cdots\!31}{69\!\cdots\!61}a^{27}-\frac{77\!\cdots\!89}{69\!\cdots\!61}a^{26}+\frac{32\!\cdots\!10}{69\!\cdots\!61}a^{25}-\frac{10\!\cdots\!88}{69\!\cdots\!61}a^{24}+\frac{32\!\cdots\!52}{69\!\cdots\!61}a^{23}-\frac{17\!\cdots\!01}{13\!\cdots\!37}a^{22}+\frac{22\!\cdots\!77}{69\!\cdots\!61}a^{21}-\frac{51\!\cdots\!07}{69\!\cdots\!61}a^{20}+\frac{11\!\cdots\!30}{69\!\cdots\!61}a^{19}-\frac{22\!\cdots\!94}{69\!\cdots\!61}a^{18}+\frac{40\!\cdots\!64}{69\!\cdots\!61}a^{17}-\frac{49\!\cdots\!03}{53\!\cdots\!97}a^{16}+\frac{82\!\cdots\!76}{63\!\cdots\!51}a^{15}-\frac{10\!\cdots\!34}{69\!\cdots\!61}a^{14}+\frac{36\!\cdots\!17}{30\!\cdots\!07}a^{13}+\frac{98\!\cdots\!69}{69\!\cdots\!61}a^{12}-\frac{97\!\cdots\!84}{53\!\cdots\!97}a^{11}+\frac{30\!\cdots\!45}{63\!\cdots\!51}a^{10}-\frac{61\!\cdots\!12}{69\!\cdots\!61}a^{9}+\frac{56\!\cdots\!50}{53\!\cdots\!97}a^{8}-\frac{78\!\cdots\!64}{69\!\cdots\!61}a^{7}+\frac{88\!\cdots\!57}{69\!\cdots\!61}a^{6}-\frac{62\!\cdots\!51}{69\!\cdots\!61}a^{5}+\frac{25\!\cdots\!27}{69\!\cdots\!61}a^{4}-\frac{10\!\cdots\!70}{61\!\cdots\!17}a^{3}+\frac{98\!\cdots\!15}{63\!\cdots\!51}a^{2}-\frac{76\!\cdots\!78}{57\!\cdots\!41}a+\frac{44\!\cdots\!13}{10\!\cdots\!97}$, $\frac{25\!\cdots\!92}{30\!\cdots\!07}a^{27}-\frac{43\!\cdots\!82}{69\!\cdots\!61}a^{26}+\frac{20\!\cdots\!33}{69\!\cdots\!61}a^{25}-\frac{74\!\cdots\!39}{69\!\cdots\!61}a^{24}+\frac{23\!\cdots\!17}{69\!\cdots\!61}a^{23}-\frac{13\!\cdots\!74}{13\!\cdots\!37}a^{22}+\frac{19\!\cdots\!63}{69\!\cdots\!61}a^{21}-\frac{47\!\cdots\!93}{69\!\cdots\!61}a^{20}+\frac{11\!\cdots\!25}{69\!\cdots\!61}a^{19}-\frac{24\!\cdots\!46}{69\!\cdots\!61}a^{18}+\frac{16\!\cdots\!73}{23\!\cdots\!39}a^{17}-\frac{71\!\cdots\!88}{53\!\cdots\!97}a^{16}+\frac{14\!\cdots\!54}{63\!\cdots\!51}a^{15}-\frac{26\!\cdots\!99}{69\!\cdots\!61}a^{14}+\frac{37\!\cdots\!93}{69\!\cdots\!61}a^{13}-\frac{50\!\cdots\!60}{69\!\cdots\!61}a^{12}+\frac{62\!\cdots\!70}{69\!\cdots\!61}a^{11}-\frac{62\!\cdots\!61}{63\!\cdots\!51}a^{10}+\frac{67\!\cdots\!07}{69\!\cdots\!61}a^{9}-\frac{48\!\cdots\!58}{53\!\cdots\!97}a^{8}+\frac{49\!\cdots\!02}{69\!\cdots\!61}a^{7}-\frac{29\!\cdots\!61}{69\!\cdots\!61}a^{6}+\frac{16\!\cdots\!18}{69\!\cdots\!61}a^{5}-\frac{94\!\cdots\!36}{69\!\cdots\!61}a^{4}+\frac{39\!\cdots\!51}{61\!\cdots\!17}a^{3}-\frac{28\!\cdots\!42}{63\!\cdots\!51}a^{2}+\frac{10\!\cdots\!09}{57\!\cdots\!41}a-\frac{38\!\cdots\!00}{10\!\cdots\!97}$, $\frac{29\!\cdots\!84}{69\!\cdots\!61}a^{27}-\frac{21\!\cdots\!95}{69\!\cdots\!61}a^{26}+\frac{96\!\cdots\!26}{69\!\cdots\!61}a^{25}-\frac{33\!\cdots\!89}{69\!\cdots\!61}a^{24}+\frac{10\!\cdots\!82}{69\!\cdots\!61}a^{23}-\frac{57\!\cdots\!50}{13\!\cdots\!37}a^{22}+\frac{79\!\cdots\!86}{69\!\cdots\!61}a^{21}-\frac{18\!\cdots\!23}{69\!\cdots\!61}a^{20}+\frac{43\!\cdots\!72}{69\!\cdots\!61}a^{19}-\frac{91\!\cdots\!41}{69\!\cdots\!61}a^{18}+\frac{17\!\cdots\!86}{69\!\cdots\!61}a^{17}-\frac{31\!\cdots\!08}{69\!\cdots\!61}a^{16}+\frac{46\!\cdots\!24}{63\!\cdots\!51}a^{15}-\frac{75\!\cdots\!19}{69\!\cdots\!61}a^{14}+\frac{98\!\cdots\!00}{69\!\cdots\!61}a^{13}-\frac{11\!\cdots\!01}{69\!\cdots\!61}a^{12}+\frac{11\!\cdots\!62}{69\!\cdots\!61}a^{11}-\frac{94\!\cdots\!76}{63\!\cdots\!51}a^{10}+\frac{51\!\cdots\!83}{69\!\cdots\!61}a^{9}-\frac{21\!\cdots\!44}{69\!\cdots\!61}a^{8}-\frac{95\!\cdots\!89}{69\!\cdots\!61}a^{7}+\frac{56\!\cdots\!43}{69\!\cdots\!61}a^{6}-\frac{52\!\cdots\!70}{69\!\cdots\!61}a^{5}-\frac{17\!\cdots\!86}{69\!\cdots\!61}a^{4}+\frac{22\!\cdots\!94}{61\!\cdots\!17}a^{3}-\frac{80\!\cdots\!96}{63\!\cdots\!51}a^{2}-\frac{17\!\cdots\!61}{57\!\cdots\!41}a-\frac{35\!\cdots\!72}{59\!\cdots\!37}$
| 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: | \( 81264745077.87146 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 81264745077.87146 \cdot 1}{2\cdot\sqrt{84980457635217037392318371738994676821454381}}\cr\approx \mathstrut & 0.658765438022473 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 8*x^27 + 39*x^26 - 147*x^25 + 484*x^24 - 1461*x^23 + 4001*x^22 - 10052*x^21 + 23841*x^20 - 52926*x^19 + 109173*x^18 - 208723*x^17 + 369896*x^16 - 608215*x^15 + 913744*x^14 - 1246890*x^13 + 1584122*x^12 - 1829692*x^11 + 1878311*x^10 - 1777347*x^9 + 1518093*x^8 - 1036936*x^7 + 598732*x^6 - 344364*x^5 + 177291*x^4 - 79332*x^3 + 46992*x^2 - 15367*x + 6413)
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^28 - 8*x^27 + 39*x^26 - 147*x^25 + 484*x^24 - 1461*x^23 + 4001*x^22 - 10052*x^21 + 23841*x^20 - 52926*x^19 + 109173*x^18 - 208723*x^17 + 369896*x^16 - 608215*x^15 + 913744*x^14 - 1246890*x^13 + 1584122*x^12 - 1829692*x^11 + 1878311*x^10 - 1777347*x^9 + 1518093*x^8 - 1036936*x^7 + 598732*x^6 - 344364*x^5 + 177291*x^4 - 79332*x^3 + 46992*x^2 - 15367*x + 6413, 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^28 - 8*x^27 + 39*x^26 - 147*x^25 + 484*x^24 - 1461*x^23 + 4001*x^22 - 10052*x^21 + 23841*x^20 - 52926*x^19 + 109173*x^18 - 208723*x^17 + 369896*x^16 - 608215*x^15 + 913744*x^14 - 1246890*x^13 + 1584122*x^12 - 1829692*x^11 + 1878311*x^10 - 1777347*x^9 + 1518093*x^8 - 1036936*x^7 + 598732*x^6 - 344364*x^5 + 177291*x^4 - 79332*x^3 + 46992*x^2 - 15367*x + 6413);
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^28 - 8*x^27 + 39*x^26 - 147*x^25 + 484*x^24 - 1461*x^23 + 4001*x^22 - 10052*x^21 + 23841*x^20 - 52926*x^19 + 109173*x^18 - 208723*x^17 + 369896*x^16 - 608215*x^15 + 913744*x^14 - 1246890*x^13 + 1584122*x^12 - 1829692*x^11 + 1878311*x^10 - 1777347*x^9 + 1518093*x^8 - 1036936*x^7 + 598732*x^6 - 344364*x^5 + 177291*x^4 - 79332*x^3 + 46992*x^2 - 15367*x + 6413);
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 56 |
| The 17 conjugacy class representatives for $D_{28}$ |
| Character table for $D_{28}$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 4.0.21901.1, 7.1.7892485271.1, 14.0.685204561282471377851.1 |
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 28 sibling: | deg 28 |
| 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 | $28$ | ${\href{/padicField/3.7.0.1}{7} }^{4}$ | ${\href{/padicField/5.14.0.1}{14} }^{2}$ | $28$ | R | ${\href{/padicField/13.2.0.1}{2} }^{14}$ | $28$ | $28$ | ${\href{/padicField/23.2.0.1}{2} }^{13}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{14}$ | ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{4}$ | $28$ | ${\href{/padicField/43.2.0.1}{2} }^{14}$ | ${\href{/padicField/47.2.0.1}{2} }^{13}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(181\)
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.181.2t1.a.a | $1$ | $ 181 $ | \(\Q(\sqrt{181}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.1991.2t1.a.a | $1$ | $ 11 \cdot 181 $ | \(\Q(\sqrt{-1991}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 2.1991.4t3.c.a | $2$ | $ 11 \cdot 181 $ | 4.2.360371.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 2.1991.14t3.a.a | $2$ | $ 11 \cdot 181 $ | 14.2.11274729599284301762821.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.1991.14t3.a.c | $2$ | $ 11 \cdot 181 $ | 14.2.11274729599284301762821.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.1991.7t2.a.b | $2$ | $ 11 \cdot 181 $ | 7.1.7892485271.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.1991.7t2.a.a | $2$ | $ 11 \cdot 181 $ | 7.1.7892485271.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.1991.14t3.a.b | $2$ | $ 11 \cdot 181 $ | 14.2.11274729599284301762821.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.1991.7t2.a.c | $2$ | $ 11 \cdot 181 $ | 7.1.7892485271.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.1991.28t10.a.e | $2$ | $ 11 \cdot 181 $ | 28.0.84980457635217037392318371738994676821454381.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.1991.28t10.a.b | $2$ | $ 11 \cdot 181 $ | 28.0.84980457635217037392318371738994676821454381.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.1991.28t10.a.a | $2$ | $ 11 \cdot 181 $ | 28.0.84980457635217037392318371738994676821454381.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.1991.28t10.a.c | $2$ | $ 11 \cdot 181 $ | 28.0.84980457635217037392318371738994676821454381.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.1991.28t10.a.f | $2$ | $ 11 \cdot 181 $ | 28.0.84980457635217037392318371738994676821454381.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.1991.28t10.a.d | $2$ | $ 11 \cdot 181 $ | 28.0.84980457635217037392318371738994676821454381.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.