Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*x + 1)
gp: K = bnfinit(y^24 - 11*y^23 + 53*y^22 - 131*y^21 + 131*y^20 + 144*y^19 - 585*y^18 + 471*y^17 + 846*y^16 - 3226*y^15 + 6073*y^14 - 7616*y^13 + 4239*y^12 + 5146*y^11 - 14303*y^10 + 15630*y^9 - 9886*y^8 + 3829*y^7 - 1007*y^6 + 240*y^5 - 15*y^4 - 39*y^3 + 19*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*x + 1)
\( x^{24} - 11 x^{23} + 53 x^{22} - 131 x^{21} + 131 x^{20} + 144 x^{19} - 585 x^{18} + 471 x^{17} + 846 x^{16} - 3226 x^{15} + 6073 x^{14} - 7616 x^{13} + 4239 x^{12} + 5146 x^{11} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(19756778413055716819205133752664064\)
\(\medspace = 2^{16}\cdot 887^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.85\) | 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))
| |
| Ramified primes: |
\(2\), \(887\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{73\!\cdots\!22}a^{23}+\frac{52\!\cdots\!67}{36\!\cdots\!61}a^{22}-\frac{12\!\cdots\!75}{73\!\cdots\!22}a^{21}+\frac{51\!\cdots\!69}{73\!\cdots\!22}a^{20}+\frac{53\!\cdots\!25}{36\!\cdots\!61}a^{19}-\frac{76\!\cdots\!73}{36\!\cdots\!61}a^{18}-\frac{12\!\cdots\!13}{73\!\cdots\!22}a^{17}-\frac{17\!\cdots\!27}{73\!\cdots\!22}a^{16}+\frac{81\!\cdots\!41}{73\!\cdots\!22}a^{15}+\frac{14\!\cdots\!03}{73\!\cdots\!22}a^{14}-\frac{30\!\cdots\!05}{73\!\cdots\!22}a^{13}+\frac{23\!\cdots\!91}{73\!\cdots\!22}a^{12}+\frac{97\!\cdots\!71}{73\!\cdots\!22}a^{11}-\frac{10\!\cdots\!01}{36\!\cdots\!61}a^{10}-\frac{13\!\cdots\!64}{36\!\cdots\!61}a^{9}+\frac{60\!\cdots\!49}{73\!\cdots\!22}a^{8}-\frac{11\!\cdots\!37}{73\!\cdots\!22}a^{7}+\frac{56\!\cdots\!01}{36\!\cdots\!61}a^{6}+\frac{18\!\cdots\!51}{73\!\cdots\!22}a^{5}-\frac{57\!\cdots\!79}{36\!\cdots\!61}a^{4}-\frac{97\!\cdots\!94}{36\!\cdots\!61}a^{3}+\frac{30\!\cdots\!24}{36\!\cdots\!61}a^{2}+\frac{66\!\cdots\!68}{36\!\cdots\!61}a-\frac{42\!\cdots\!50}{36\!\cdots\!61}$
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: | $13$ | 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: |
$\frac{15\!\cdots\!41}{73\!\cdots\!22}a^{23}-\frac{26\!\cdots\!37}{73\!\cdots\!22}a^{22}+\frac{17\!\cdots\!03}{73\!\cdots\!22}a^{21}-\frac{54\!\cdots\!35}{73\!\cdots\!22}a^{20}+\frac{37\!\cdots\!33}{36\!\cdots\!61}a^{19}+\frac{39\!\cdots\!63}{73\!\cdots\!22}a^{18}-\frac{29\!\cdots\!45}{73\!\cdots\!22}a^{17}+\frac{31\!\cdots\!27}{73\!\cdots\!22}a^{16}+\frac{16\!\cdots\!28}{36\!\cdots\!61}a^{15}-\frac{13\!\cdots\!45}{73\!\cdots\!22}a^{14}+\frac{12\!\cdots\!49}{36\!\cdots\!61}a^{13}-\frac{16\!\cdots\!02}{36\!\cdots\!61}a^{12}+\frac{92\!\cdots\!42}{36\!\cdots\!61}a^{11}+\frac{28\!\cdots\!55}{73\!\cdots\!22}a^{10}-\frac{36\!\cdots\!93}{36\!\cdots\!61}a^{9}+\frac{67\!\cdots\!95}{73\!\cdots\!22}a^{8}-\frac{23\!\cdots\!47}{73\!\cdots\!22}a^{7}-\frac{57\!\cdots\!97}{73\!\cdots\!22}a^{6}+\frac{28\!\cdots\!65}{36\!\cdots\!61}a^{5}-\frac{92\!\cdots\!95}{73\!\cdots\!22}a^{4}+\frac{23\!\cdots\!91}{73\!\cdots\!22}a^{3}-\frac{41\!\cdots\!23}{73\!\cdots\!22}a^{2}+\frac{14\!\cdots\!57}{73\!\cdots\!22}a+\frac{30\!\cdots\!41}{36\!\cdots\!61}$, $\frac{18\!\cdots\!15}{73\!\cdots\!22}a^{23}-\frac{95\!\cdots\!59}{36\!\cdots\!61}a^{22}+\frac{43\!\cdots\!62}{36\!\cdots\!61}a^{21}-\frac{96\!\cdots\!35}{36\!\cdots\!61}a^{20}+\frac{13\!\cdots\!11}{73\!\cdots\!22}a^{19}+\frac{33\!\cdots\!65}{73\!\cdots\!22}a^{18}-\frac{44\!\cdots\!41}{36\!\cdots\!61}a^{17}+\frac{18\!\cdots\!10}{36\!\cdots\!61}a^{16}+\frac{17\!\cdots\!53}{73\!\cdots\!22}a^{15}-\frac{49\!\cdots\!31}{73\!\cdots\!22}a^{14}+\frac{84\!\cdots\!97}{73\!\cdots\!22}a^{13}-\frac{47\!\cdots\!88}{36\!\cdots\!61}a^{12}+\frac{13\!\cdots\!87}{36\!\cdots\!61}a^{11}+\frac{10\!\cdots\!67}{73\!\cdots\!22}a^{10}-\frac{19\!\cdots\!87}{73\!\cdots\!22}a^{9}+\frac{17\!\cdots\!05}{73\!\cdots\!22}a^{8}-\frac{45\!\cdots\!76}{36\!\cdots\!61}a^{7}+\frac{14\!\cdots\!52}{36\!\cdots\!61}a^{6}-\frac{72\!\cdots\!91}{73\!\cdots\!22}a^{5}+\frac{38\!\cdots\!21}{73\!\cdots\!22}a^{4}+\frac{66\!\cdots\!89}{36\!\cdots\!61}a^{3}-\frac{24\!\cdots\!57}{36\!\cdots\!61}a^{2}+\frac{89\!\cdots\!31}{36\!\cdots\!61}a-\frac{89\!\cdots\!03}{73\!\cdots\!22}$, $\frac{81\!\cdots\!63}{36\!\cdots\!61}a^{23}-\frac{79\!\cdots\!29}{36\!\cdots\!61}a^{22}+\frac{33\!\cdots\!20}{36\!\cdots\!61}a^{21}-\frac{62\!\cdots\!57}{36\!\cdots\!61}a^{20}+\frac{12\!\cdots\!05}{36\!\cdots\!61}a^{19}+\frac{17\!\cdots\!00}{36\!\cdots\!61}a^{18}-\frac{29\!\cdots\!22}{36\!\cdots\!61}a^{17}-\frac{39\!\cdots\!25}{36\!\cdots\!61}a^{16}+\frac{82\!\cdots\!37}{36\!\cdots\!61}a^{15}-\frac{17\!\cdots\!72}{36\!\cdots\!61}a^{14}+\frac{25\!\cdots\!49}{36\!\cdots\!61}a^{13}-\frac{21\!\cdots\!68}{36\!\cdots\!61}a^{12}-\frac{97\!\cdots\!23}{36\!\cdots\!61}a^{11}+\frac{52\!\cdots\!80}{36\!\cdots\!61}a^{10}-\frac{62\!\cdots\!42}{36\!\cdots\!61}a^{9}+\frac{31\!\cdots\!38}{36\!\cdots\!61}a^{8}+\frac{17\!\cdots\!77}{36\!\cdots\!61}a^{7}-\frac{10\!\cdots\!37}{36\!\cdots\!61}a^{6}+\frac{48\!\cdots\!18}{36\!\cdots\!61}a^{5}-\frac{92\!\cdots\!37}{36\!\cdots\!61}a^{4}+\frac{10\!\cdots\!69}{36\!\cdots\!61}a^{3}-\frac{62\!\cdots\!67}{36\!\cdots\!61}a^{2}-\frac{88\!\cdots\!83}{36\!\cdots\!61}a+\frac{23\!\cdots\!09}{36\!\cdots\!61}$, $\frac{18\!\cdots\!15}{73\!\cdots\!22}a^{23}-\frac{95\!\cdots\!59}{36\!\cdots\!61}a^{22}+\frac{43\!\cdots\!62}{36\!\cdots\!61}a^{21}-\frac{96\!\cdots\!35}{36\!\cdots\!61}a^{20}+\frac{13\!\cdots\!11}{73\!\cdots\!22}a^{19}+\frac{33\!\cdots\!65}{73\!\cdots\!22}a^{18}-\frac{44\!\cdots\!41}{36\!\cdots\!61}a^{17}+\frac{18\!\cdots\!10}{36\!\cdots\!61}a^{16}+\frac{17\!\cdots\!53}{73\!\cdots\!22}a^{15}-\frac{49\!\cdots\!31}{73\!\cdots\!22}a^{14}+\frac{84\!\cdots\!97}{73\!\cdots\!22}a^{13}-\frac{47\!\cdots\!88}{36\!\cdots\!61}a^{12}+\frac{13\!\cdots\!87}{36\!\cdots\!61}a^{11}+\frac{10\!\cdots\!67}{73\!\cdots\!22}a^{10}-\frac{19\!\cdots\!87}{73\!\cdots\!22}a^{9}+\frac{17\!\cdots\!05}{73\!\cdots\!22}a^{8}-\frac{45\!\cdots\!76}{36\!\cdots\!61}a^{7}+\frac{14\!\cdots\!52}{36\!\cdots\!61}a^{6}-\frac{72\!\cdots\!91}{73\!\cdots\!22}a^{5}+\frac{38\!\cdots\!21}{73\!\cdots\!22}a^{4}+\frac{66\!\cdots\!89}{36\!\cdots\!61}a^{3}-\frac{24\!\cdots\!57}{36\!\cdots\!61}a^{2}+\frac{89\!\cdots\!31}{36\!\cdots\!61}a-\frac{16\!\cdots\!81}{73\!\cdots\!22}$, $a$, $\frac{31\!\cdots\!86}{36\!\cdots\!61}a^{23}-\frac{65\!\cdots\!35}{73\!\cdots\!22}a^{22}+\frac{29\!\cdots\!99}{73\!\cdots\!22}a^{21}-\frac{62\!\cdots\!57}{73\!\cdots\!22}a^{20}+\frac{39\!\cdots\!45}{73\!\cdots\!22}a^{19}+\frac{58\!\cdots\!01}{36\!\cdots\!61}a^{18}-\frac{28\!\cdots\!83}{73\!\cdots\!22}a^{17}+\frac{99\!\cdots\!11}{73\!\cdots\!22}a^{16}+\frac{59\!\cdots\!03}{73\!\cdots\!22}a^{15}-\frac{81\!\cdots\!11}{36\!\cdots\!61}a^{14}+\frac{27\!\cdots\!65}{73\!\cdots\!22}a^{13}-\frac{14\!\cdots\!42}{36\!\cdots\!61}a^{12}+\frac{34\!\cdots\!00}{36\!\cdots\!61}a^{11}+\frac{18\!\cdots\!69}{36\!\cdots\!61}a^{10}-\frac{64\!\cdots\!45}{73\!\cdots\!22}a^{9}+\frac{27\!\cdots\!51}{36\!\cdots\!61}a^{8}-\frac{25\!\cdots\!95}{73\!\cdots\!22}a^{7}+\frac{74\!\cdots\!45}{73\!\cdots\!22}a^{6}-\frac{19\!\cdots\!63}{73\!\cdots\!22}a^{5}+\frac{16\!\cdots\!54}{36\!\cdots\!61}a^{4}+\frac{23\!\cdots\!07}{73\!\cdots\!22}a^{3}-\frac{14\!\cdots\!79}{73\!\cdots\!22}a^{2}+\frac{21\!\cdots\!03}{73\!\cdots\!22}a-\frac{15\!\cdots\!75}{73\!\cdots\!22}$, $\frac{78\!\cdots\!04}{36\!\cdots\!61}a^{23}-\frac{68\!\cdots\!06}{36\!\cdots\!61}a^{22}+\frac{23\!\cdots\!51}{36\!\cdots\!61}a^{21}-\frac{24\!\cdots\!47}{36\!\cdots\!61}a^{20}-\frac{65\!\cdots\!48}{36\!\cdots\!61}a^{19}+\frac{42\!\cdots\!17}{73\!\cdots\!22}a^{18}-\frac{13\!\cdots\!37}{36\!\cdots\!61}a^{17}-\frac{79\!\cdots\!71}{73\!\cdots\!22}a^{16}+\frac{18\!\cdots\!19}{73\!\cdots\!22}a^{15}-\frac{89\!\cdots\!14}{36\!\cdots\!61}a^{14}+\frac{39\!\cdots\!48}{36\!\cdots\!61}a^{13}+\frac{26\!\cdots\!81}{73\!\cdots\!22}a^{12}-\frac{89\!\cdots\!79}{73\!\cdots\!22}a^{11}+\frac{11\!\cdots\!35}{73\!\cdots\!22}a^{10}-\frac{24\!\cdots\!93}{73\!\cdots\!22}a^{9}-\frac{10\!\cdots\!13}{73\!\cdots\!22}a^{8}+\frac{13\!\cdots\!83}{73\!\cdots\!22}a^{7}-\frac{73\!\cdots\!07}{73\!\cdots\!22}a^{6}+\frac{10\!\cdots\!42}{36\!\cdots\!61}a^{5}-\frac{29\!\cdots\!66}{36\!\cdots\!61}a^{4}+\frac{26\!\cdots\!07}{73\!\cdots\!22}a^{3}-\frac{32\!\cdots\!53}{73\!\cdots\!22}a^{2}-\frac{11\!\cdots\!39}{36\!\cdots\!61}a-\frac{22\!\cdots\!31}{73\!\cdots\!22}$, $\frac{17\!\cdots\!55}{36\!\cdots\!61}a^{23}-\frac{39\!\cdots\!53}{73\!\cdots\!22}a^{22}+\frac{19\!\cdots\!81}{73\!\cdots\!22}a^{21}-\frac{47\!\cdots\!55}{73\!\cdots\!22}a^{20}+\frac{47\!\cdots\!11}{73\!\cdots\!22}a^{19}+\frac{54\!\cdots\!11}{73\!\cdots\!22}a^{18}-\frac{21\!\cdots\!41}{73\!\cdots\!22}a^{17}+\frac{87\!\cdots\!26}{36\!\cdots\!61}a^{16}+\frac{15\!\cdots\!20}{36\!\cdots\!61}a^{15}-\frac{59\!\cdots\!54}{36\!\cdots\!61}a^{14}+\frac{22\!\cdots\!21}{73\!\cdots\!22}a^{13}-\frac{27\!\cdots\!35}{73\!\cdots\!22}a^{12}+\frac{14\!\cdots\!15}{73\!\cdots\!22}a^{11}+\frac{20\!\cdots\!99}{73\!\cdots\!22}a^{10}-\frac{26\!\cdots\!76}{36\!\cdots\!61}a^{9}+\frac{56\!\cdots\!75}{73\!\cdots\!22}a^{8}-\frac{16\!\cdots\!58}{36\!\cdots\!61}a^{7}+\frac{53\!\cdots\!72}{36\!\cdots\!61}a^{6}-\frac{16\!\cdots\!53}{73\!\cdots\!22}a^{5}+\frac{37\!\cdots\!36}{36\!\cdots\!61}a^{4}+\frac{94\!\cdots\!54}{36\!\cdots\!61}a^{3}-\frac{83\!\cdots\!89}{36\!\cdots\!61}a^{2}+\frac{46\!\cdots\!35}{73\!\cdots\!22}a-\frac{66\!\cdots\!50}{36\!\cdots\!61}$, $\frac{22\!\cdots\!55}{36\!\cdots\!61}a^{23}-\frac{23\!\cdots\!24}{36\!\cdots\!61}a^{22}+\frac{10\!\cdots\!94}{36\!\cdots\!61}a^{21}-\frac{25\!\cdots\!25}{36\!\cdots\!61}a^{20}+\frac{43\!\cdots\!47}{73\!\cdots\!22}a^{19}+\frac{34\!\cdots\!77}{36\!\cdots\!61}a^{18}-\frac{22\!\cdots\!93}{73\!\cdots\!22}a^{17}+\frac{14\!\cdots\!55}{73\!\cdots\!22}a^{16}+\frac{18\!\cdots\!48}{36\!\cdots\!61}a^{15}-\frac{63\!\cdots\!54}{36\!\cdots\!61}a^{14}+\frac{23\!\cdots\!27}{73\!\cdots\!22}a^{13}-\frac{27\!\cdots\!37}{73\!\cdots\!22}a^{12}+\frac{12\!\cdots\!45}{73\!\cdots\!22}a^{11}+\frac{22\!\cdots\!31}{73\!\cdots\!22}a^{10}-\frac{53\!\cdots\!67}{73\!\cdots\!22}a^{9}+\frac{55\!\cdots\!01}{73\!\cdots\!22}a^{8}-\frac{33\!\cdots\!71}{73\!\cdots\!22}a^{7}+\frac{64\!\cdots\!02}{36\!\cdots\!61}a^{6}-\frac{16\!\cdots\!77}{36\!\cdots\!61}a^{5}+\frac{56\!\cdots\!67}{73\!\cdots\!22}a^{4}+\frac{10\!\cdots\!49}{73\!\cdots\!22}a^{3}-\frac{86\!\cdots\!75}{36\!\cdots\!61}a^{2}+\frac{75\!\cdots\!33}{73\!\cdots\!22}a-\frac{10\!\cdots\!12}{36\!\cdots\!61}$, $\frac{10\!\cdots\!31}{36\!\cdots\!61}a^{23}-\frac{20\!\cdots\!19}{73\!\cdots\!22}a^{22}+\frac{86\!\cdots\!39}{73\!\cdots\!22}a^{21}-\frac{88\!\cdots\!57}{36\!\cdots\!61}a^{20}+\frac{87\!\cdots\!05}{73\!\cdots\!22}a^{19}+\frac{18\!\cdots\!14}{36\!\cdots\!61}a^{18}-\frac{39\!\cdots\!31}{36\!\cdots\!61}a^{17}+\frac{73\!\cdots\!29}{36\!\cdots\!61}a^{16}+\frac{89\!\cdots\!70}{36\!\cdots\!61}a^{15}-\frac{45\!\cdots\!23}{73\!\cdots\!22}a^{14}+\frac{38\!\cdots\!14}{36\!\cdots\!61}a^{13}-\frac{39\!\cdots\!98}{36\!\cdots\!61}a^{12}+\frac{61\!\cdots\!14}{36\!\cdots\!61}a^{11}+\frac{52\!\cdots\!09}{36\!\cdots\!61}a^{10}-\frac{17\!\cdots\!77}{73\!\cdots\!22}a^{9}+\frac{71\!\cdots\!38}{36\!\cdots\!61}a^{8}-\frac{37\!\cdots\!19}{36\!\cdots\!61}a^{7}+\frac{14\!\cdots\!40}{36\!\cdots\!61}a^{6}-\frac{51\!\cdots\!60}{36\!\cdots\!61}a^{5}+\frac{14\!\cdots\!73}{73\!\cdots\!22}a^{4}+\frac{55\!\cdots\!79}{36\!\cdots\!61}a^{3}-\frac{77\!\cdots\!55}{73\!\cdots\!22}a^{2}+\frac{20\!\cdots\!01}{73\!\cdots\!22}a-\frac{34\!\cdots\!14}{36\!\cdots\!61}$, $\frac{10\!\cdots\!02}{36\!\cdots\!61}a^{23}-\frac{12\!\cdots\!72}{36\!\cdots\!61}a^{22}+\frac{13\!\cdots\!51}{73\!\cdots\!22}a^{21}-\frac{18\!\cdots\!74}{36\!\cdots\!61}a^{20}+\frac{25\!\cdots\!25}{36\!\cdots\!61}a^{19}+\frac{10\!\cdots\!41}{73\!\cdots\!22}a^{18}-\frac{16\!\cdots\!25}{73\!\cdots\!22}a^{17}+\frac{21\!\cdots\!49}{73\!\cdots\!22}a^{16}+\frac{11\!\cdots\!39}{73\!\cdots\!22}a^{15}-\frac{88\!\cdots\!05}{73\!\cdots\!22}a^{14}+\frac{94\!\cdots\!00}{36\!\cdots\!61}a^{13}-\frac{13\!\cdots\!45}{36\!\cdots\!61}a^{12}+\frac{10\!\cdots\!79}{36\!\cdots\!61}a^{11}+\frac{33\!\cdots\!51}{36\!\cdots\!61}a^{10}-\frac{21\!\cdots\!93}{36\!\cdots\!61}a^{9}+\frac{58\!\cdots\!89}{73\!\cdots\!22}a^{8}-\frac{43\!\cdots\!67}{73\!\cdots\!22}a^{7}+\frac{19\!\cdots\!97}{73\!\cdots\!22}a^{6}-\frac{47\!\cdots\!55}{73\!\cdots\!22}a^{5}+\frac{73\!\cdots\!23}{73\!\cdots\!22}a^{4}-\frac{18\!\cdots\!07}{36\!\cdots\!61}a^{3}-\frac{74\!\cdots\!97}{36\!\cdots\!61}a^{2}+\frac{94\!\cdots\!83}{73\!\cdots\!22}a-\frac{72\!\cdots\!52}{36\!\cdots\!61}$, $\frac{30\!\cdots\!95}{73\!\cdots\!22}a^{23}-\frac{44\!\cdots\!49}{73\!\cdots\!22}a^{22}+\frac{26\!\cdots\!97}{73\!\cdots\!22}a^{21}-\frac{83\!\cdots\!59}{73\!\cdots\!22}a^{20}+\frac{60\!\cdots\!94}{36\!\cdots\!61}a^{19}+\frac{13\!\cdots\!25}{36\!\cdots\!61}a^{18}-\frac{40\!\cdots\!39}{73\!\cdots\!22}a^{17}+\frac{26\!\cdots\!67}{36\!\cdots\!61}a^{16}+\frac{31\!\cdots\!05}{73\!\cdots\!22}a^{15}-\frac{20\!\cdots\!31}{73\!\cdots\!22}a^{14}+\frac{20\!\cdots\!04}{36\!\cdots\!61}a^{13}-\frac{56\!\cdots\!79}{73\!\cdots\!22}a^{12}+\frac{41\!\cdots\!11}{73\!\cdots\!22}a^{11}+\frac{13\!\cdots\!69}{36\!\cdots\!61}a^{10}-\frac{10\!\cdots\!41}{73\!\cdots\!22}a^{9}+\frac{62\!\cdots\!33}{36\!\cdots\!61}a^{8}-\frac{36\!\cdots\!44}{36\!\cdots\!61}a^{7}+\frac{84\!\cdots\!12}{36\!\cdots\!61}a^{6}+\frac{23\!\cdots\!53}{36\!\cdots\!61}a^{5}+\frac{18\!\cdots\!15}{73\!\cdots\!22}a^{4}+\frac{14\!\cdots\!67}{36\!\cdots\!61}a^{3}-\frac{40\!\cdots\!30}{36\!\cdots\!61}a^{2}+\frac{12\!\cdots\!15}{73\!\cdots\!22}a-\frac{83\!\cdots\!91}{73\!\cdots\!22}$, $\frac{19\!\cdots\!22}{36\!\cdots\!61}a^{23}-\frac{43\!\cdots\!47}{73\!\cdots\!22}a^{22}+\frac{10\!\cdots\!29}{36\!\cdots\!61}a^{21}-\frac{57\!\cdots\!57}{73\!\cdots\!22}a^{20}+\frac{39\!\cdots\!26}{36\!\cdots\!61}a^{19}+\frac{27\!\cdots\!67}{73\!\cdots\!22}a^{18}-\frac{20\!\cdots\!15}{73\!\cdots\!22}a^{17}+\frac{31\!\cdots\!57}{73\!\cdots\!22}a^{16}+\frac{35\!\cdots\!51}{36\!\cdots\!61}a^{15}-\frac{12\!\cdots\!29}{73\!\cdots\!22}a^{14}+\frac{15\!\cdots\!38}{36\!\cdots\!61}a^{13}-\frac{44\!\cdots\!25}{73\!\cdots\!22}a^{12}+\frac{39\!\cdots\!71}{73\!\cdots\!22}a^{11}-\frac{20\!\cdots\!81}{73\!\cdots\!22}a^{10}-\frac{29\!\cdots\!73}{36\!\cdots\!61}a^{9}+\frac{50\!\cdots\!96}{36\!\cdots\!61}a^{8}-\frac{94\!\cdots\!25}{73\!\cdots\!22}a^{7}+\frac{26\!\cdots\!86}{36\!\cdots\!61}a^{6}-\frac{73\!\cdots\!22}{36\!\cdots\!61}a^{5}-\frac{19\!\cdots\!19}{36\!\cdots\!61}a^{4}+\frac{76\!\cdots\!53}{36\!\cdots\!61}a^{3}-\frac{64\!\cdots\!57}{73\!\cdots\!22}a^{2}+\frac{17\!\cdots\!25}{73\!\cdots\!22}a+\frac{13\!\cdots\!77}{73\!\cdots\!22}$
| 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: | \( 33313220.242250312 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 33313220.242250312 \cdot 1}{2\cdot\sqrt{19756778413055716819205133752664064}}\cr\approx \mathstrut & 0.181822331396691 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*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^24 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*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^24 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*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^24 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*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
$\SL(2,5):C_2$ (as 24T576):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A non-solvable group of order 240 |
| The 18 conjugacy class representatives for $\SL(2,5):C_2$ |
| Character table for $\SL(2,5):C_2$ |
Intermediate fields
| 6.2.12588304.1, 12.4.158465397596416.1 |
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 40 siblings: | data not computed |
| Arithmetically equvalently sibling: | 24.4.19756778413055716819205133752664064.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.3.0.1}{3} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{8}$ | $20{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }^{6}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | $20{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ |
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
|
\(887\)
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $2$ | $4$ | $4$ | ||||
| Deg $8$ | $2$ | $4$ | $4$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.887.2t1.a.a | $1$ | $ 887 $ | \(\Q(\sqrt{-887}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.3548.120.b.a | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.3548.120.b.b | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.3548.120.b.c | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.3548.120.b.d | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 3.3147076.12t33.a.a | $3$ | $ 2^{2} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 3.3147076.12t33.a.b | $3$ | $ 2^{2} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| * | 3.3548.12t76.a.a | $3$ | $ 2^{2} \cdot 887 $ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| * | 3.3548.12t76.a.b | $3$ | $ 2^{2} \cdot 887 $ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| 4.3147076.10t11.a.a | $4$ | $ 2^{2} \cdot 887^{2}$ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $0$ | |
| 4.3147076.5t4.a.a | $4$ | $ 2^{2} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $0$ | |
| 4.3147076.40t188.b.a | $4$ | $ 2^{2} \cdot 887^{2}$ | 24.4.19756778413055716819205133752664064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 4.3147076.40t188.b.b | $4$ | $ 2^{2} \cdot 887^{2}$ | 24.4.19756778413055716819205133752664064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 5.11165825648.12t75.a.a | $5$ | $ 2^{4} \cdot 887^{3}$ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $-1$ | |
| * | 5.12588304.6t12.a.a | $5$ | $ 2^{4} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $1$ |
| * | 6.11165825648.24t576.b.a | $6$ | $ 2^{4} \cdot 887^{3}$ | 24.4.19756778413055716819205133752664064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |
| * | 6.11165825648.24t576.b.b | $6$ | $ 2^{4} \cdot 887^{3}$ | 24.4.19756778413055716819205133752664064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |
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 *.