Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 3*x^22 + x^21 + 7*x^20 + 20*x^19 + 16*x^18 - 5*x^17 + 80*x^16 + 120*x^15 - 277*x^14 + 191*x^13 + 511*x^12 - 356*x^11 - 47*x^10 + 255*x^9 - 75*x^8 - 130*x^7 + 96*x^6 + 20*x^5 - 43*x^4 + 14*x^3 + 3*x^2 - 4*x + 1)
gp: K = bnfinit(y^24 - y^23 + 3*y^22 + y^21 + 7*y^20 + 20*y^19 + 16*y^18 - 5*y^17 + 80*y^16 + 120*y^15 - 277*y^14 + 191*y^13 + 511*y^12 - 356*y^11 - 47*y^10 + 255*y^9 - 75*y^8 - 130*y^7 + 96*y^6 + 20*y^5 - 43*y^4 + 14*y^3 + 3*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 + 3*x^22 + x^21 + 7*x^20 + 20*x^19 + 16*x^18 - 5*x^17 + 80*x^16 + 120*x^15 - 277*x^14 + 191*x^13 + 511*x^12 - 356*x^11 - 47*x^10 + 255*x^9 - 75*x^8 - 130*x^7 + 96*x^6 + 20*x^5 - 43*x^4 + 14*x^3 + 3*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 + 3*x^22 + x^21 + 7*x^20 + 20*x^19 + 16*x^18 - 5*x^17 + 80*x^16 + 120*x^15 - 277*x^14 + 191*x^13 + 511*x^12 - 356*x^11 - 47*x^10 + 255*x^9 - 75*x^8 - 130*x^7 + 96*x^6 + 20*x^5 - 43*x^4 + 14*x^3 + 3*x^2 - 4*x + 1)
\( x^{24} - x^{23} + 3 x^{22} + x^{21} + 7 x^{20} + 20 x^{19} + 16 x^{18} - 5 x^{17} + 80 x^{16} + 120 x^{15} + \cdots + 1 \)
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: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(43070843460234840091705322265625\)
\(\medspace = 5^{18}\cdot 7^{12}\cdot 13^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.80\) | 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: | $5^{3/4}7^{1/2}13^{2/3}\approx 48.91087415715005$ | ||
| Ramified primes: |
\(5\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $12$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5}a^{20}+\frac{2}{5}a^{19}-\frac{2}{5}a^{15}-\frac{2}{5}a^{13}-\frac{1}{5}a^{10}+\frac{2}{5}a^{7}+\frac{2}{5}a^{5}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{25}a^{21}+\frac{11}{25}a^{19}+\frac{2}{5}a^{18}-\frac{7}{25}a^{16}-\frac{11}{25}a^{15}+\frac{3}{25}a^{14}+\frac{9}{25}a^{13}+\frac{2}{5}a^{12}-\frac{1}{25}a^{11}+\frac{12}{25}a^{10}-\frac{1}{5}a^{9}+\frac{12}{25}a^{8}+\frac{6}{25}a^{7}-\frac{8}{25}a^{6}-\frac{9}{25}a^{5}-\frac{2}{5}a^{3}-\frac{7}{25}a^{2}-\frac{1}{5}a-\frac{2}{25}$, $\frac{1}{48579775}a^{22}+\frac{929372}{48579775}a^{21}+\frac{3117066}{48579775}a^{20}-\frac{24048838}{48579775}a^{19}-\frac{1365806}{9715955}a^{18}+\frac{21188693}{48579775}a^{17}-\frac{2297098}{9715955}a^{16}+\frac{9906676}{48579775}a^{15}+\frac{898481}{1943191}a^{14}+\frac{19022948}{48579775}a^{13}-\frac{10156881}{48579775}a^{12}+\frac{2825313}{9715955}a^{11}-\frac{20718171}{48579775}a^{10}-\frac{21859023}{48579775}a^{9}-\frac{3245356}{9715955}a^{8}+\frac{14409459}{48579775}a^{7}-\frac{2074297}{9715955}a^{6}-\frac{23899363}{48579775}a^{5}-\frac{2957667}{9715955}a^{4}+\frac{20623173}{48579775}a^{3}-\frac{20094159}{48579775}a^{2}-\frac{23026697}{48579775}a+\frac{1108186}{48579775}$, $\frac{1}{61\!\cdots\!25}a^{23}+\frac{3891036209}{61\!\cdots\!25}a^{22}+\frac{11\!\cdots\!79}{61\!\cdots\!25}a^{21}+\frac{21\!\cdots\!54}{61\!\cdots\!25}a^{20}+\frac{14\!\cdots\!53}{61\!\cdots\!25}a^{19}-\frac{60\!\cdots\!52}{61\!\cdots\!25}a^{18}-\frac{18\!\cdots\!49}{61\!\cdots\!25}a^{17}-\frac{21\!\cdots\!22}{61\!\cdots\!25}a^{16}-\frac{27\!\cdots\!52}{61\!\cdots\!25}a^{15}-\frac{49\!\cdots\!61}{12\!\cdots\!05}a^{14}-\frac{75\!\cdots\!39}{61\!\cdots\!25}a^{13}+\frac{22\!\cdots\!83}{61\!\cdots\!25}a^{12}+\frac{22\!\cdots\!47}{12\!\cdots\!05}a^{11}+\frac{82\!\cdots\!63}{61\!\cdots\!25}a^{10}+\frac{22\!\cdots\!99}{61\!\cdots\!25}a^{9}+\frac{25\!\cdots\!12}{61\!\cdots\!25}a^{8}+\frac{27\!\cdots\!92}{61\!\cdots\!25}a^{7}+\frac{17\!\cdots\!58}{24\!\cdots\!81}a^{6}-\frac{14\!\cdots\!32}{61\!\cdots\!25}a^{5}-\frac{28\!\cdots\!22}{61\!\cdots\!25}a^{4}+\frac{27\!\cdots\!27}{61\!\cdots\!25}a^{3}-\frac{26\!\cdots\!23}{61\!\cdots\!25}a^{2}-\frac{15\!\cdots\!73}{61\!\cdots\!25}a+\frac{16\!\cdots\!34}{61\!\cdots\!25}$
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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{1457897166849640138}{1237089868843836905} a^{23} + \frac{639520975469448258}{1237089868843836905} a^{22} - \frac{756379969124315973}{247417973768767381} a^{21} - \frac{3858059012597590842}{1237089868843836905} a^{20} - \frac{11523994884991911402}{1237089868843836905} a^{19} - \frac{35650695603798790534}{1237089868843836905} a^{18} - \frac{41390760923071746673}{1237089868843836905} a^{17} - \frac{11605642499717166083}{1237089868843836905} a^{16} - \frac{119480271356447902036}{1237089868843836905} a^{15} - \frac{241992695946705777594}{1237089868843836905} a^{14} + \frac{57671368561121023977}{247417973768767381} a^{13} - \frac{92608538521353306109}{1237089868843836905} a^{12} - \frac{858213682288180728847}{1237089868843836905} a^{11} + \frac{20901943470918526935}{247417973768767381} a^{10} + \frac{209116674464092941903}{1237089868843836905} a^{9} - \frac{332864747274619755427}{1237089868843836905} a^{8} - \frac{33487518560139406134}{1237089868843836905} a^{7} + \frac{196869994074594769109}{1237089868843836905} a^{6} - \frac{46453301483365639537}{1237089868843836905} a^{5} - \frac{72706267410129211654}{1237089868843836905} a^{4} + \frac{42637110868590830109}{1237089868843836905} a^{3} - \frac{649116610022975024}{247417973768767381} a^{2} - \frac{8900225595830356236}{1237089868843836905} a + \frac{4761840995787026854}{1237089868843836905} \)
(order $10$)
| 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: |
$\frac{46\!\cdots\!44}{24\!\cdots\!81}a^{23}-\frac{27\!\cdots\!64}{24\!\cdots\!81}a^{22}+\frac{67\!\cdots\!84}{12\!\cdots\!05}a^{21}+\frac{94\!\cdots\!20}{24\!\cdots\!81}a^{20}+\frac{19\!\cdots\!74}{12\!\cdots\!05}a^{19}+\frac{10\!\cdots\!36}{24\!\cdots\!81}a^{18}+\frac{12\!\cdots\!97}{24\!\cdots\!81}a^{17}+\frac{18\!\cdots\!77}{12\!\cdots\!05}a^{16}+\frac{19\!\cdots\!86}{12\!\cdots\!05}a^{15}+\frac{35\!\cdots\!32}{12\!\cdots\!05}a^{14}-\frac{47\!\cdots\!39}{12\!\cdots\!05}a^{13}+\frac{57\!\cdots\!79}{24\!\cdots\!81}a^{12}+\frac{12\!\cdots\!61}{12\!\cdots\!05}a^{11}-\frac{28\!\cdots\!57}{12\!\cdots\!05}a^{10}-\frac{17\!\cdots\!09}{24\!\cdots\!81}a^{9}+\frac{48\!\cdots\!93}{12\!\cdots\!05}a^{8}+\frac{97\!\cdots\!94}{12\!\cdots\!05}a^{7}-\frac{25\!\cdots\!87}{12\!\cdots\!05}a^{6}+\frac{11\!\cdots\!69}{12\!\cdots\!05}a^{5}+\frac{12\!\cdots\!02}{24\!\cdots\!81}a^{4}-\frac{11\!\cdots\!99}{24\!\cdots\!81}a^{3}+\frac{12\!\cdots\!32}{12\!\cdots\!05}a^{2}+\frac{13\!\cdots\!52}{24\!\cdots\!81}a-\frac{55\!\cdots\!73}{12\!\cdots\!05}$, $\frac{15\!\cdots\!58}{61\!\cdots\!25}a^{23}-\frac{79\!\cdots\!57}{61\!\cdots\!25}a^{22}+\frac{40\!\cdots\!09}{61\!\cdots\!25}a^{21}+\frac{36\!\cdots\!38}{61\!\cdots\!25}a^{20}+\frac{12\!\cdots\!56}{61\!\cdots\!25}a^{19}+\frac{36\!\cdots\!04}{61\!\cdots\!25}a^{18}+\frac{41\!\cdots\!11}{61\!\cdots\!25}a^{17}+\frac{10\!\cdots\!79}{61\!\cdots\!25}a^{16}+\frac{50\!\cdots\!09}{24\!\cdots\!81}a^{15}+\frac{48\!\cdots\!36}{12\!\cdots\!05}a^{14}-\frac{31\!\cdots\!09}{61\!\cdots\!25}a^{13}+\frac{12\!\cdots\!63}{61\!\cdots\!25}a^{12}+\frac{17\!\cdots\!71}{12\!\cdots\!05}a^{11}-\frac{14\!\cdots\!52}{61\!\cdots\!25}a^{10}-\frac{19\!\cdots\!41}{61\!\cdots\!25}a^{9}+\frac{32\!\cdots\!46}{61\!\cdots\!25}a^{8}+\frac{95\!\cdots\!51}{12\!\cdots\!05}a^{7}-\frac{38\!\cdots\!46}{12\!\cdots\!05}a^{6}+\frac{56\!\cdots\!51}{61\!\cdots\!25}a^{5}+\frac{69\!\cdots\!39}{61\!\cdots\!25}a^{4}-\frac{39\!\cdots\!26}{61\!\cdots\!25}a^{3}+\frac{42\!\cdots\!97}{61\!\cdots\!25}a^{2}+\frac{71\!\cdots\!64}{61\!\cdots\!25}a-\frac{30\!\cdots\!82}{61\!\cdots\!25}$, $\frac{92\!\cdots\!61}{61\!\cdots\!25}a^{23}+\frac{48\!\cdots\!72}{12\!\cdots\!05}a^{22}-\frac{77\!\cdots\!49}{61\!\cdots\!25}a^{21}+\frac{37\!\cdots\!29}{12\!\cdots\!05}a^{20}-\frac{24\!\cdots\!54}{12\!\cdots\!05}a^{19}+\frac{38\!\cdots\!73}{61\!\cdots\!25}a^{18}+\frac{31\!\cdots\!34}{61\!\cdots\!25}a^{17}-\frac{84\!\cdots\!62}{61\!\cdots\!25}a^{16}-\frac{40\!\cdots\!01}{61\!\cdots\!25}a^{15}+\frac{87\!\cdots\!73}{12\!\cdots\!05}a^{14}-\frac{39\!\cdots\!91}{61\!\cdots\!25}a^{13}-\frac{15\!\cdots\!03}{61\!\cdots\!25}a^{12}+\frac{63\!\cdots\!52}{12\!\cdots\!05}a^{11}-\frac{67\!\cdots\!58}{61\!\cdots\!25}a^{10}-\frac{45\!\cdots\!64}{61\!\cdots\!25}a^{9}+\frac{25\!\cdots\!07}{61\!\cdots\!25}a^{8}+\frac{10\!\cdots\!26}{61\!\cdots\!25}a^{7}-\frac{27\!\cdots\!66}{12\!\cdots\!05}a^{6}+\frac{17\!\cdots\!64}{12\!\cdots\!05}a^{5}+\frac{90\!\cdots\!98}{61\!\cdots\!25}a^{4}-\frac{24\!\cdots\!38}{24\!\cdots\!81}a^{3}-\frac{10\!\cdots\!82}{61\!\cdots\!25}a^{2}+\frac{29\!\cdots\!91}{12\!\cdots\!05}a-\frac{11\!\cdots\!27}{12\!\cdots\!05}$, $\frac{15\!\cdots\!93}{61\!\cdots\!25}a^{23}-\frac{67\!\cdots\!86}{61\!\cdots\!25}a^{22}+\frac{42\!\cdots\!24}{61\!\cdots\!25}a^{21}+\frac{37\!\cdots\!19}{61\!\cdots\!25}a^{20}+\frac{12\!\cdots\!71}{61\!\cdots\!25}a^{19}+\frac{37\!\cdots\!34}{61\!\cdots\!25}a^{18}+\frac{45\!\cdots\!79}{61\!\cdots\!25}a^{17}+\frac{19\!\cdots\!43}{61\!\cdots\!25}a^{16}+\frac{13\!\cdots\!73}{61\!\cdots\!25}a^{15}+\frac{25\!\cdots\!59}{61\!\cdots\!25}a^{14}-\frac{27\!\cdots\!89}{61\!\cdots\!25}a^{13}+\frac{14\!\cdots\!12}{61\!\cdots\!25}a^{12}+\frac{82\!\cdots\!27}{61\!\cdots\!25}a^{11}-\frac{56\!\cdots\!27}{61\!\cdots\!25}a^{10}-\frac{63\!\cdots\!54}{61\!\cdots\!25}a^{9}+\frac{30\!\cdots\!02}{61\!\cdots\!25}a^{8}+\frac{68\!\cdots\!52}{61\!\cdots\!25}a^{7}-\frac{14\!\cdots\!59}{61\!\cdots\!25}a^{6}+\frac{53\!\cdots\!06}{61\!\cdots\!25}a^{5}+\frac{53\!\cdots\!09}{61\!\cdots\!25}a^{4}-\frac{26\!\cdots\!98}{61\!\cdots\!25}a^{3}+\frac{46\!\cdots\!87}{61\!\cdots\!25}a^{2}+\frac{46\!\cdots\!47}{61\!\cdots\!25}a-\frac{15\!\cdots\!17}{61\!\cdots\!25}$, $\frac{64\!\cdots\!56}{61\!\cdots\!25}a^{23}-\frac{12\!\cdots\!53}{12\!\cdots\!05}a^{22}+\frac{19\!\cdots\!44}{61\!\cdots\!25}a^{21}+\frac{13\!\cdots\!17}{12\!\cdots\!05}a^{20}+\frac{46\!\cdots\!38}{61\!\cdots\!25}a^{19}+\frac{13\!\cdots\!63}{61\!\cdots\!25}a^{18}+\frac{11\!\cdots\!89}{61\!\cdots\!25}a^{17}-\frac{15\!\cdots\!38}{61\!\cdots\!25}a^{16}+\frac{52\!\cdots\!46}{61\!\cdots\!25}a^{15}+\frac{79\!\cdots\!34}{61\!\cdots\!25}a^{14}-\frac{17\!\cdots\!34}{61\!\cdots\!25}a^{13}+\frac{12\!\cdots\!67}{61\!\cdots\!25}a^{12}+\frac{32\!\cdots\!62}{61\!\cdots\!25}a^{11}-\frac{21\!\cdots\!82}{61\!\cdots\!25}a^{10}-\frac{93\!\cdots\!09}{61\!\cdots\!25}a^{9}+\frac{16\!\cdots\!98}{61\!\cdots\!25}a^{8}-\frac{44\!\cdots\!11}{61\!\cdots\!25}a^{7}-\frac{79\!\cdots\!89}{61\!\cdots\!25}a^{6}+\frac{54\!\cdots\!43}{61\!\cdots\!25}a^{5}+\frac{11\!\cdots\!83}{61\!\cdots\!25}a^{4}-\frac{52\!\cdots\!56}{12\!\cdots\!05}a^{3}+\frac{83\!\cdots\!42}{61\!\cdots\!25}a^{2}+\frac{18\!\cdots\!67}{12\!\cdots\!05}a-\frac{13\!\cdots\!66}{61\!\cdots\!25}$, $\frac{19\!\cdots\!76}{12\!\cdots\!05}a^{23}-\frac{43\!\cdots\!98}{61\!\cdots\!25}a^{22}+\frac{25\!\cdots\!38}{61\!\cdots\!25}a^{21}+\frac{24\!\cdots\!42}{61\!\cdots\!25}a^{20}+\frac{77\!\cdots\!48}{61\!\cdots\!25}a^{19}+\frac{46\!\cdots\!29}{12\!\cdots\!05}a^{18}+\frac{27\!\cdots\!56}{61\!\cdots\!25}a^{17}+\frac{84\!\cdots\!87}{61\!\cdots\!25}a^{16}+\frac{79\!\cdots\!48}{61\!\cdots\!25}a^{15}+\frac{15\!\cdots\!42}{61\!\cdots\!25}a^{14}-\frac{18\!\cdots\!78}{61\!\cdots\!25}a^{13}+\frac{71\!\cdots\!63}{61\!\cdots\!25}a^{12}+\frac{54\!\cdots\!66}{61\!\cdots\!25}a^{11}-\frac{63\!\cdots\!24}{61\!\cdots\!25}a^{10}-\frac{10\!\cdots\!86}{61\!\cdots\!25}a^{9}+\frac{20\!\cdots\!18}{61\!\cdots\!25}a^{8}+\frac{19\!\cdots\!27}{61\!\cdots\!25}a^{7}-\frac{12\!\cdots\!32}{61\!\cdots\!25}a^{6}+\frac{31\!\cdots\!43}{61\!\cdots\!25}a^{5}+\frac{81\!\cdots\!69}{12\!\cdots\!05}a^{4}-\frac{26\!\cdots\!94}{61\!\cdots\!25}a^{3}+\frac{14\!\cdots\!74}{61\!\cdots\!25}a^{2}+\frac{42\!\cdots\!66}{61\!\cdots\!25}a-\frac{24\!\cdots\!71}{61\!\cdots\!25}$, $\frac{54\!\cdots\!07}{61\!\cdots\!25}a^{23}-\frac{32\!\cdots\!91}{61\!\cdots\!25}a^{22}+\frac{13\!\cdots\!93}{61\!\cdots\!25}a^{21}+\frac{12\!\cdots\!84}{61\!\cdots\!25}a^{20}+\frac{37\!\cdots\!71}{61\!\cdots\!25}a^{19}+\frac{12\!\cdots\!36}{61\!\cdots\!25}a^{18}+\frac{24\!\cdots\!42}{12\!\cdots\!05}a^{17}-\frac{21\!\cdots\!88}{12\!\cdots\!05}a^{16}+\frac{40\!\cdots\!34}{61\!\cdots\!25}a^{15}+\frac{82\!\cdots\!74}{61\!\cdots\!25}a^{14}-\frac{13\!\cdots\!03}{61\!\cdots\!25}a^{13}+\frac{59\!\cdots\!12}{12\!\cdots\!05}a^{12}+\frac{33\!\cdots\!32}{61\!\cdots\!25}a^{11}-\frac{96\!\cdots\!84}{61\!\cdots\!25}a^{10}-\frac{28\!\cdots\!06}{12\!\cdots\!05}a^{9}+\frac{27\!\cdots\!18}{12\!\cdots\!05}a^{8}+\frac{82\!\cdots\!66}{61\!\cdots\!25}a^{7}-\frac{10\!\cdots\!34}{61\!\cdots\!25}a^{6}+\frac{18\!\cdots\!66}{61\!\cdots\!25}a^{5}+\frac{39\!\cdots\!11}{61\!\cdots\!25}a^{4}-\frac{21\!\cdots\!58}{61\!\cdots\!25}a^{3}-\frac{34\!\cdots\!71}{61\!\cdots\!25}a^{2}+\frac{51\!\cdots\!47}{61\!\cdots\!25}a-\frac{15\!\cdots\!42}{61\!\cdots\!25}$, $\frac{12\!\cdots\!89}{12\!\cdots\!05}a^{23}-\frac{42\!\cdots\!12}{61\!\cdots\!25}a^{22}+\frac{40\!\cdots\!28}{61\!\cdots\!25}a^{21}-\frac{94\!\cdots\!77}{61\!\cdots\!25}a^{20}-\frac{29\!\cdots\!97}{61\!\cdots\!25}a^{19}-\frac{32\!\cdots\!47}{12\!\cdots\!05}a^{18}-\frac{73\!\cdots\!86}{61\!\cdots\!25}a^{17}-\frac{95\!\cdots\!39}{61\!\cdots\!25}a^{16}+\frac{33\!\cdots\!21}{61\!\cdots\!25}a^{15}-\frac{21\!\cdots\!84}{61\!\cdots\!25}a^{14}-\frac{73\!\cdots\!28}{61\!\cdots\!25}a^{13}+\frac{90\!\cdots\!32}{61\!\cdots\!25}a^{12}-\frac{26\!\cdots\!27}{61\!\cdots\!25}a^{11}-\frac{22\!\cdots\!84}{61\!\cdots\!25}a^{10}+\frac{36\!\cdots\!11}{61\!\cdots\!25}a^{9}+\frac{58\!\cdots\!64}{61\!\cdots\!25}a^{8}-\frac{65\!\cdots\!76}{61\!\cdots\!25}a^{7}-\frac{21\!\cdots\!31}{61\!\cdots\!25}a^{6}+\frac{50\!\cdots\!88}{61\!\cdots\!25}a^{5}-\frac{13\!\cdots\!19}{12\!\cdots\!05}a^{4}-\frac{16\!\cdots\!71}{61\!\cdots\!25}a^{3}+\frac{79\!\cdots\!14}{61\!\cdots\!25}a^{2}+\frac{97\!\cdots\!49}{61\!\cdots\!25}a-\frac{10\!\cdots\!61}{61\!\cdots\!25}$, $\frac{70\!\cdots\!94}{61\!\cdots\!25}a^{23}-\frac{34\!\cdots\!27}{61\!\cdots\!25}a^{22}+\frac{15\!\cdots\!59}{61\!\cdots\!25}a^{21}+\frac{27\!\cdots\!43}{61\!\cdots\!25}a^{20}+\frac{11\!\cdots\!06}{12\!\cdots\!05}a^{19}+\frac{19\!\cdots\!67}{61\!\cdots\!25}a^{18}+\frac{51\!\cdots\!61}{12\!\cdots\!05}a^{17}+\frac{11\!\cdots\!29}{61\!\cdots\!25}a^{16}+\frac{11\!\cdots\!18}{12\!\cdots\!05}a^{15}+\frac{14\!\cdots\!22}{61\!\cdots\!25}a^{14}-\frac{10\!\cdots\!89}{61\!\cdots\!25}a^{13}-\frac{38\!\cdots\!09}{12\!\cdots\!05}a^{12}+\frac{47\!\cdots\!56}{61\!\cdots\!25}a^{11}+\frac{71\!\cdots\!63}{61\!\cdots\!25}a^{10}-\frac{33\!\cdots\!37}{12\!\cdots\!05}a^{9}+\frac{18\!\cdots\!36}{61\!\cdots\!25}a^{8}+\frac{13\!\cdots\!79}{12\!\cdots\!05}a^{7}-\frac{10\!\cdots\!07}{61\!\cdots\!25}a^{6}+\frac{57\!\cdots\!72}{12\!\cdots\!05}a^{5}+\frac{57\!\cdots\!37}{61\!\cdots\!25}a^{4}-\frac{22\!\cdots\!06}{61\!\cdots\!25}a^{3}-\frac{37\!\cdots\!03}{61\!\cdots\!25}a^{2}+\frac{70\!\cdots\!04}{61\!\cdots\!25}a-\frac{94\!\cdots\!48}{24\!\cdots\!81}$, $\frac{11\!\cdots\!28}{61\!\cdots\!25}a^{23}+\frac{91\!\cdots\!31}{61\!\cdots\!25}a^{22}+\frac{22\!\cdots\!26}{61\!\cdots\!25}a^{21}+\frac{70\!\cdots\!01}{61\!\cdots\!25}a^{20}+\frac{12\!\cdots\!43}{61\!\cdots\!25}a^{19}+\frac{40\!\cdots\!09}{61\!\cdots\!25}a^{18}+\frac{26\!\cdots\!90}{24\!\cdots\!81}a^{17}+\frac{45\!\cdots\!42}{61\!\cdots\!25}a^{16}+\frac{10\!\cdots\!12}{61\!\cdots\!25}a^{15}+\frac{31\!\cdots\!63}{61\!\cdots\!25}a^{14}-\frac{13\!\cdots\!66}{61\!\cdots\!25}a^{13}-\frac{47\!\cdots\!68}{12\!\cdots\!05}a^{12}+\frac{84\!\cdots\!89}{61\!\cdots\!25}a^{11}+\frac{74\!\cdots\!67}{61\!\cdots\!25}a^{10}-\frac{14\!\cdots\!59}{24\!\cdots\!81}a^{9}+\frac{82\!\cdots\!53}{61\!\cdots\!25}a^{8}+\frac{33\!\cdots\!68}{61\!\cdots\!25}a^{7}-\frac{82\!\cdots\!98}{61\!\cdots\!25}a^{6}-\frac{10\!\cdots\!82}{61\!\cdots\!25}a^{5}+\frac{48\!\cdots\!69}{61\!\cdots\!25}a^{4}+\frac{28\!\cdots\!88}{61\!\cdots\!25}a^{3}-\frac{21\!\cdots\!62}{61\!\cdots\!25}a^{2}+\frac{31\!\cdots\!63}{61\!\cdots\!25}a+\frac{23\!\cdots\!19}{61\!\cdots\!25}$, $\frac{57\!\cdots\!03}{61\!\cdots\!25}a^{23}+\frac{85\!\cdots\!58}{12\!\cdots\!05}a^{22}-\frac{16\!\cdots\!85}{24\!\cdots\!81}a^{21}+\frac{28\!\cdots\!17}{12\!\cdots\!05}a^{20}+\frac{30\!\cdots\!92}{61\!\cdots\!25}a^{19}+\frac{35\!\cdots\!74}{61\!\cdots\!25}a^{18}+\frac{86\!\cdots\!37}{61\!\cdots\!25}a^{17}+\frac{30\!\cdots\!53}{24\!\cdots\!81}a^{16}-\frac{88\!\cdots\!49}{12\!\cdots\!05}a^{15}+\frac{36\!\cdots\!66}{61\!\cdots\!25}a^{14}+\frac{97\!\cdots\!44}{12\!\cdots\!05}a^{13}-\frac{11\!\cdots\!59}{61\!\cdots\!25}a^{12}+\frac{96\!\cdots\!73}{61\!\cdots\!25}a^{11}+\frac{73\!\cdots\!34}{24\!\cdots\!81}a^{10}-\frac{12\!\cdots\!37}{61\!\cdots\!25}a^{9}+\frac{48\!\cdots\!24}{12\!\cdots\!05}a^{8}+\frac{11\!\cdots\!02}{12\!\cdots\!05}a^{7}-\frac{28\!\cdots\!71}{61\!\cdots\!25}a^{6}-\frac{42\!\cdots\!03}{61\!\cdots\!25}a^{5}+\frac{29\!\cdots\!54}{61\!\cdots\!25}a^{4}-\frac{42\!\cdots\!08}{12\!\cdots\!05}a^{3}-\frac{28\!\cdots\!92}{24\!\cdots\!81}a^{2}+\frac{84\!\cdots\!71}{12\!\cdots\!05}a-\frac{73\!\cdots\!94}{61\!\cdots\!25}$
| 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: | \( 3136393.8562446074 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 3136393.8562446074 \cdot 1}{10\cdot\sqrt{43070843460234840091705322265625}}\cr\approx \mathstrut & 0.180924477225979 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 3*x^22 + x^21 + 7*x^20 + 20*x^19 + 16*x^18 - 5*x^17 + 80*x^16 + 120*x^15 - 277*x^14 + 191*x^13 + 511*x^12 - 356*x^11 - 47*x^10 + 255*x^9 - 75*x^8 - 130*x^7 + 96*x^6 + 20*x^5 - 43*x^4 + 14*x^3 + 3*x^2 - 4*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 - x^23 + 3*x^22 + x^21 + 7*x^20 + 20*x^19 + 16*x^18 - 5*x^17 + 80*x^16 + 120*x^15 - 277*x^14 + 191*x^13 + 511*x^12 - 356*x^11 - 47*x^10 + 255*x^9 - 75*x^8 - 130*x^7 + 96*x^6 + 20*x^5 - 43*x^4 + 14*x^3 + 3*x^2 - 4*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 - x^23 + 3*x^22 + x^21 + 7*x^20 + 20*x^19 + 16*x^18 - 5*x^17 + 80*x^16 + 120*x^15 - 277*x^14 + 191*x^13 + 511*x^12 - 356*x^11 - 47*x^10 + 255*x^9 - 75*x^8 - 130*x^7 + 96*x^6 + 20*x^5 - 43*x^4 + 14*x^3 + 3*x^2 - 4*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 - x^23 + 3*x^22 + x^21 + 7*x^20 + 20*x^19 + 16*x^18 - 5*x^17 + 80*x^16 + 120*x^15 - 277*x^14 + 191*x^13 + 511*x^12 - 356*x^11 - 47*x^10 + 255*x^9 - 75*x^8 - 130*x^7 + 96*x^6 + 20*x^5 - 43*x^4 + 14*x^3 + 3*x^2 - 4*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
$S_3\times C_{12}$ (as 24T65):
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 72 |
| The 36 conjugacy class representatives for $S_3\times C_{12}$ |
| Character table for $S_3\times C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-35}) \), 4.4.6125.1, \(\Q(\zeta_{5})\), \(\Q(\sqrt{5}, \sqrt{-7})\), 6.0.7245875.1, 8.0.37515625.1, 12.0.52502704515625.1 |
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 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 | ${\href{/padicField/2.12.0.1}{12} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{3}$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{8}$ | R | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{12}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(7\)
| Deg $24$ | $2$ | $12$ | $12$ | |||
|
\(13\)
| 13.12.8.3 | $x^{12} - 78 x^{9} + 2197 x^{6} + 290004 x^{3} + 114244$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
| 13.12.0.1 | $x^{12} + x^{8} + 5 x^{7} + 8 x^{6} + 11 x^{5} + 3 x^{4} + x^{3} + x^{2} + 4 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.91.6t1.g.a | $1$ | $ 7 \cdot 13 $ | 6.0.9796423.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.13.3t1.a.a | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.91.6t1.g.b | $1$ | $ 7 \cdot 13 $ | 6.0.9796423.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.65.6t1.b.a | $1$ | $ 5 \cdot 13 $ | 6.6.3570125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.455.6t1.b.a | $1$ | $ 5 \cdot 7 \cdot 13 $ | 6.0.1224552875.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.65.6t1.b.b | $1$ | $ 5 \cdot 13 $ | 6.6.3570125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.13.3t1.a.b | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.455.6t1.b.b | $1$ | $ 5 \cdot 7 \cdot 13 $ | 6.0.1224552875.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.35.4t1.a.a | $1$ | $ 5 \cdot 7 $ | 4.4.6125.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.35.4t1.a.b | $1$ | $ 5 \cdot 7 $ | 4.4.6125.1 | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.455.12t1.a.a | $1$ | $ 5 \cdot 7 \cdot 13 $ | 12.12.187441217958845703125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.455.12t1.a.b | $1$ | $ 5 \cdot 7 \cdot 13 $ | 12.12.187441217958845703125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.455.12t1.a.c | $1$ | $ 5 \cdot 7 \cdot 13 $ | 12.12.187441217958845703125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.455.12t1.a.d | $1$ | $ 5 \cdot 7 \cdot 13 $ | 12.12.187441217958845703125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.65.12t1.a.a | $1$ | $ 5 \cdot 13 $ | 12.0.1593224064453125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.65.12t1.a.b | $1$ | $ 5 \cdot 13 $ | 12.0.1593224064453125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.65.12t1.a.c | $1$ | $ 5 \cdot 13 $ | 12.0.1593224064453125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.65.12t1.a.d | $1$ | $ 5 \cdot 13 $ | 12.0.1593224064453125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.5915.3t2.a.a | $2$ | $ 5 \cdot 7 \cdot 13^{2}$ | 3.1.5915.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.5915.6t3.f.a | $2$ | $ 5 \cdot 7 \cdot 13^{2}$ | 6.2.174936125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.455.12t18.a.a | $2$ | $ 5 \cdot 7 \cdot 13 $ | 12.0.52502704515625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.455.6t5.a.a | $2$ | $ 5 \cdot 7 \cdot 13 $ | 6.0.7245875.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.455.12t18.a.b | $2$ | $ 5 \cdot 7 \cdot 13 $ | 12.0.52502704515625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.455.6t5.a.b | $2$ | $ 5 \cdot 7 \cdot 13 $ | 6.0.7245875.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| 2.29575.12t11.b.a | $2$ | $ 5^{2} \cdot 7 \cdot 13^{2}$ | 12.4.187441217958845703125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.29575.12t11.b.b | $2$ | $ 5^{2} \cdot 7 \cdot 13^{2}$ | 12.4.187441217958845703125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.2275.24t65.a.a | $2$ | $ 5^{2} \cdot 7 \cdot 13 $ | 24.0.43070843460234840091705322265625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2275.24t65.a.b | $2$ | $ 5^{2} \cdot 7 \cdot 13 $ | 24.0.43070843460234840091705322265625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2275.24t65.a.c | $2$ | $ 5^{2} \cdot 7 \cdot 13 $ | 24.0.43070843460234840091705322265625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2275.24t65.a.d | $2$ | $ 5^{2} \cdot 7 \cdot 13 $ | 24.0.43070843460234840091705322265625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
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 *.