Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 16*x^22 - 26*x^21 + 31*x^20 - 48*x^19 + 128*x^18 - 293*x^17 + 478*x^16 - 623*x^15 + 874*x^14 - 1534*x^13 + 2419*x^12 - 2374*x^11 + 149*x^10 + 3878*x^9 - 7423*x^8 + 8318*x^7 - 6533*x^6 + 3783*x^5 - 1636*x^4 + 521*x^3 - 116*x^2 + 16*x - 1)
gp: K = bnfinit(y^24 - 6*y^23 + 16*y^22 - 26*y^21 + 31*y^20 - 48*y^19 + 128*y^18 - 293*y^17 + 478*y^16 - 623*y^15 + 874*y^14 - 1534*y^13 + 2419*y^12 - 2374*y^11 + 149*y^10 + 3878*y^9 - 7423*y^8 + 8318*y^7 - 6533*y^6 + 3783*y^5 - 1636*y^4 + 521*y^3 - 116*y^2 + 16*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 6*x^23 + 16*x^22 - 26*x^21 + 31*x^20 - 48*x^19 + 128*x^18 - 293*x^17 + 478*x^16 - 623*x^15 + 874*x^14 - 1534*x^13 + 2419*x^12 - 2374*x^11 + 149*x^10 + 3878*x^9 - 7423*x^8 + 8318*x^7 - 6533*x^6 + 3783*x^5 - 1636*x^4 + 521*x^3 - 116*x^2 + 16*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^23 + 16*x^22 - 26*x^21 + 31*x^20 - 48*x^19 + 128*x^18 - 293*x^17 + 478*x^16 - 623*x^15 + 874*x^14 - 1534*x^13 + 2419*x^12 - 2374*x^11 + 149*x^10 + 3878*x^9 - 7423*x^8 + 8318*x^7 - 6533*x^6 + 3783*x^5 - 1636*x^4 + 521*x^3 - 116*x^2 + 16*x - 1)
\( x^{24} - 6 x^{23} + 16 x^{22} - 26 x^{21} + 31 x^{20} - 48 x^{19} + 128 x^{18} - 293 x^{17} + 478 x^{16} + \cdots - 1 \)
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: |
\(641953627807088196277618408203125\)
\(\medspace = 5^{31}\cdot 13^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.28\) | 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: | $5^{31/20}13^{1/2}\approx 43.68930970521314$ | ||
| Ramified primes: |
\(5\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{65}a^{22}-\frac{23}{65}a^{21}+\frac{2}{5}a^{20}-\frac{5}{13}a^{19}-\frac{5}{13}a^{18}+\frac{22}{65}a^{17}-\frac{16}{65}a^{16}-\frac{18}{65}a^{15}-\frac{2}{13}a^{14}-\frac{6}{13}a^{13}-\frac{6}{65}a^{12}-\frac{12}{65}a^{11}-\frac{31}{65}a^{10}-\frac{1}{13}a^{9}+\frac{4}{13}a^{8}-\frac{17}{65}a^{7}+\frac{1}{65}a^{6}+\frac{23}{65}a^{5}-\frac{5}{13}a^{4}-\frac{1}{13}a^{3}-\frac{21}{65}a^{2}-\frac{12}{65}a+\frac{29}{65}$, $\frac{1}{11\!\cdots\!55}a^{23}-\frac{24178171201512}{11\!\cdots\!55}a^{22}+\frac{765180508217198}{11\!\cdots\!55}a^{21}-\frac{256457269528524}{11\!\cdots\!55}a^{20}-\frac{360384211274816}{23\!\cdots\!91}a^{19}+\frac{24\!\cdots\!27}{11\!\cdots\!55}a^{18}-\frac{48\!\cdots\!34}{11\!\cdots\!55}a^{17}+\frac{163302506738507}{910756351016035}a^{16}+\frac{50\!\cdots\!17}{11\!\cdots\!55}a^{15}-\frac{11\!\cdots\!03}{23\!\cdots\!91}a^{14}-\frac{28\!\cdots\!86}{11\!\cdots\!55}a^{13}+\frac{17\!\cdots\!87}{11\!\cdots\!55}a^{12}-\frac{627005720087303}{11\!\cdots\!55}a^{11}-\frac{56\!\cdots\!86}{11\!\cdots\!55}a^{10}+\frac{730403218181634}{23\!\cdots\!91}a^{9}-\frac{209070435330582}{11\!\cdots\!55}a^{8}-\frac{25\!\cdots\!81}{11\!\cdots\!55}a^{7}+\frac{53\!\cdots\!34}{11\!\cdots\!55}a^{6}+\frac{54\!\cdots\!78}{11\!\cdots\!55}a^{5}+\frac{612816325047007}{23\!\cdots\!91}a^{4}+\frac{23\!\cdots\!09}{11\!\cdots\!55}a^{3}+\frac{50\!\cdots\!92}{11\!\cdots\!55}a^{2}-\frac{47\!\cdots\!33}{11\!\cdots\!55}a+\frac{48\!\cdots\!29}{11\!\cdots\!55}$
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: |
$\frac{12\!\cdots\!86}{11\!\cdots\!55}a^{23}-\frac{78\!\cdots\!92}{11\!\cdots\!55}a^{22}+\frac{20\!\cdots\!63}{11\!\cdots\!55}a^{21}-\frac{32\!\cdots\!09}{11\!\cdots\!55}a^{20}+\frac{74\!\cdots\!75}{23\!\cdots\!91}a^{19}-\frac{44\!\cdots\!81}{910756351016035}a^{18}+\frac{16\!\cdots\!86}{11\!\cdots\!55}a^{17}-\frac{37\!\cdots\!59}{11\!\cdots\!55}a^{16}+\frac{59\!\cdots\!37}{11\!\cdots\!55}a^{15}-\frac{15\!\cdots\!75}{23\!\cdots\!91}a^{14}+\frac{10\!\cdots\!99}{11\!\cdots\!55}a^{13}-\frac{19\!\cdots\!98}{11\!\cdots\!55}a^{12}+\frac{30\!\cdots\!82}{11\!\cdots\!55}a^{11}-\frac{28\!\cdots\!86}{11\!\cdots\!55}a^{10}-\frac{38\!\cdots\!56}{182151270203207}a^{9}+\frac{54\!\cdots\!38}{11\!\cdots\!55}a^{8}-\frac{94\!\cdots\!06}{11\!\cdots\!55}a^{7}+\frac{99\!\cdots\!04}{11\!\cdots\!55}a^{6}-\frac{72\!\cdots\!12}{11\!\cdots\!55}a^{5}+\frac{76\!\cdots\!28}{23\!\cdots\!91}a^{4}-\frac{14\!\cdots\!11}{11\!\cdots\!55}a^{3}+\frac{39\!\cdots\!52}{11\!\cdots\!55}a^{2}-\frac{67\!\cdots\!53}{11\!\cdots\!55}a+\frac{53\!\cdots\!29}{11\!\cdots\!55}$, $\frac{13\!\cdots\!94}{11\!\cdots\!55}a^{23}-\frac{74\!\cdots\!73}{11\!\cdots\!55}a^{22}+\frac{18\!\cdots\!17}{11\!\cdots\!55}a^{21}-\frac{26\!\cdots\!51}{11\!\cdots\!55}a^{20}+\frac{58\!\cdots\!72}{23\!\cdots\!91}a^{19}-\frac{51\!\cdots\!72}{11\!\cdots\!55}a^{18}+\frac{14\!\cdots\!29}{11\!\cdots\!55}a^{17}-\frac{32\!\cdots\!21}{11\!\cdots\!55}a^{16}+\frac{49\!\cdots\!63}{11\!\cdots\!55}a^{15}-\frac{12\!\cdots\!34}{23\!\cdots\!91}a^{14}+\frac{89\!\cdots\!46}{11\!\cdots\!55}a^{13}-\frac{16\!\cdots\!27}{11\!\cdots\!55}a^{12}+\frac{24\!\cdots\!98}{11\!\cdots\!55}a^{11}-\frac{20\!\cdots\!14}{11\!\cdots\!55}a^{10}-\frac{15\!\cdots\!97}{23\!\cdots\!91}a^{9}+\frac{48\!\cdots\!57}{11\!\cdots\!55}a^{8}-\frac{77\!\cdots\!09}{11\!\cdots\!55}a^{7}+\frac{58\!\cdots\!57}{910756351016035}a^{6}-\frac{52\!\cdots\!58}{11\!\cdots\!55}a^{5}+\frac{52\!\cdots\!25}{23\!\cdots\!91}a^{4}-\frac{96\!\cdots\!39}{11\!\cdots\!55}a^{3}+\frac{24\!\cdots\!93}{11\!\cdots\!55}a^{2}-\frac{38\!\cdots\!32}{11\!\cdots\!55}a+\frac{27\!\cdots\!36}{11\!\cdots\!55}$, $\frac{27\!\cdots\!47}{11\!\cdots\!55}a^{23}+\frac{14\!\cdots\!29}{11\!\cdots\!55}a^{22}-\frac{36\!\cdots\!96}{11\!\cdots\!55}a^{21}+\frac{52\!\cdots\!43}{11\!\cdots\!55}a^{20}-\frac{11\!\cdots\!53}{23\!\cdots\!91}a^{19}+\frac{10\!\cdots\!51}{11\!\cdots\!55}a^{18}-\frac{29\!\cdots\!37}{11\!\cdots\!55}a^{17}+\frac{64\!\cdots\!63}{11\!\cdots\!55}a^{16}-\frac{97\!\cdots\!24}{11\!\cdots\!55}a^{15}+\frac{24\!\cdots\!24}{23\!\cdots\!91}a^{14}-\frac{17\!\cdots\!28}{11\!\cdots\!55}a^{13}+\frac{32\!\cdots\!16}{11\!\cdots\!55}a^{12}-\frac{49\!\cdots\!04}{11\!\cdots\!55}a^{11}+\frac{40\!\cdots\!72}{11\!\cdots\!55}a^{10}+\frac{31\!\cdots\!31}{23\!\cdots\!91}a^{9}-\frac{97\!\cdots\!96}{11\!\cdots\!55}a^{8}+\frac{15\!\cdots\!42}{11\!\cdots\!55}a^{7}-\frac{15\!\cdots\!13}{11\!\cdots\!55}a^{6}+\frac{10\!\cdots\!79}{11\!\cdots\!55}a^{5}-\frac{10\!\cdots\!56}{23\!\cdots\!91}a^{4}+\frac{18\!\cdots\!37}{11\!\cdots\!55}a^{3}-\frac{48\!\cdots\!89}{11\!\cdots\!55}a^{2}+\frac{76\!\cdots\!56}{11\!\cdots\!55}a-\frac{53\!\cdots\!28}{11\!\cdots\!55}$, $a$, $\frac{91\!\cdots\!11}{910756351016035}a^{23}-\frac{62\!\cdots\!21}{11\!\cdots\!55}a^{22}+\frac{14\!\cdots\!94}{11\!\cdots\!55}a^{21}-\frac{15\!\cdots\!89}{910756351016035}a^{20}+\frac{44\!\cdots\!66}{23\!\cdots\!91}a^{19}-\frac{41\!\cdots\!84}{11\!\cdots\!55}a^{18}+\frac{12\!\cdots\!63}{11\!\cdots\!55}a^{17}-\frac{26\!\cdots\!57}{11\!\cdots\!55}a^{16}+\frac{38\!\cdots\!41}{11\!\cdots\!55}a^{15}-\frac{93\!\cdots\!52}{23\!\cdots\!91}a^{14}+\frac{70\!\cdots\!52}{11\!\cdots\!55}a^{13}-\frac{13\!\cdots\!84}{11\!\cdots\!55}a^{12}+\frac{19\!\cdots\!91}{11\!\cdots\!55}a^{11}-\frac{14\!\cdots\!93}{11\!\cdots\!55}a^{10}-\frac{17\!\cdots\!61}{23\!\cdots\!91}a^{9}+\frac{40\!\cdots\!94}{11\!\cdots\!55}a^{8}-\frac{59\!\cdots\!38}{11\!\cdots\!55}a^{7}+\frac{56\!\cdots\!57}{11\!\cdots\!55}a^{6}-\frac{37\!\cdots\!06}{11\!\cdots\!55}a^{5}+\frac{36\!\cdots\!55}{23\!\cdots\!91}a^{4}-\frac{64\!\cdots\!53}{11\!\cdots\!55}a^{3}+\frac{16\!\cdots\!21}{11\!\cdots\!55}a^{2}-\frac{25\!\cdots\!54}{11\!\cdots\!55}a+\frac{19\!\cdots\!37}{11\!\cdots\!55}$, $\frac{31\!\cdots\!69}{11\!\cdots\!55}a^{23}+\frac{17\!\cdots\!31}{11\!\cdots\!55}a^{22}-\frac{42\!\cdots\!66}{11\!\cdots\!55}a^{21}+\frac{62\!\cdots\!69}{11\!\cdots\!55}a^{20}-\frac{13\!\cdots\!38}{23\!\cdots\!91}a^{19}+\frac{11\!\cdots\!22}{11\!\cdots\!55}a^{18}-\frac{27\!\cdots\!16}{910756351016035}a^{17}+\frac{77\!\cdots\!63}{11\!\cdots\!55}a^{16}-\frac{11\!\cdots\!32}{11\!\cdots\!55}a^{15}+\frac{28\!\cdots\!60}{23\!\cdots\!91}a^{14}-\frac{21\!\cdots\!41}{11\!\cdots\!55}a^{13}+\frac{39\!\cdots\!39}{11\!\cdots\!55}a^{12}-\frac{58\!\cdots\!44}{11\!\cdots\!55}a^{11}+\frac{48\!\cdots\!51}{11\!\cdots\!55}a^{10}+\frac{36\!\cdots\!47}{23\!\cdots\!91}a^{9}-\frac{11\!\cdots\!47}{11\!\cdots\!55}a^{8}+\frac{18\!\cdots\!43}{11\!\cdots\!55}a^{7}-\frac{17\!\cdots\!33}{11\!\cdots\!55}a^{6}+\frac{94\!\cdots\!04}{910756351016035}a^{5}-\frac{12\!\cdots\!93}{23\!\cdots\!91}a^{4}+\frac{22\!\cdots\!64}{11\!\cdots\!55}a^{3}-\frac{57\!\cdots\!31}{11\!\cdots\!55}a^{2}+\frac{92\!\cdots\!16}{11\!\cdots\!55}a-\frac{68\!\cdots\!49}{11\!\cdots\!55}$, $\frac{17\!\cdots\!86}{11\!\cdots\!55}a^{23}-\frac{98\!\cdots\!93}{11\!\cdots\!55}a^{22}+\frac{18\!\cdots\!57}{910756351016035}a^{21}-\frac{67\!\cdots\!24}{23\!\cdots\!91}a^{20}+\frac{74\!\cdots\!00}{23\!\cdots\!91}a^{19}-\frac{65\!\cdots\!98}{11\!\cdots\!55}a^{18}+\frac{19\!\cdots\!54}{11\!\cdots\!55}a^{17}-\frac{42\!\cdots\!13}{11\!\cdots\!55}a^{16}+\frac{12\!\cdots\!09}{23\!\cdots\!91}a^{15}-\frac{15\!\cdots\!47}{23\!\cdots\!91}a^{14}+\frac{11\!\cdots\!34}{11\!\cdots\!55}a^{13}-\frac{21\!\cdots\!72}{11\!\cdots\!55}a^{12}+\frac{31\!\cdots\!89}{11\!\cdots\!55}a^{11}-\frac{50\!\cdots\!07}{23\!\cdots\!91}a^{10}-\frac{22\!\cdots\!80}{23\!\cdots\!91}a^{9}+\frac{63\!\cdots\!03}{11\!\cdots\!55}a^{8}-\frac{98\!\cdots\!34}{11\!\cdots\!55}a^{7}+\frac{95\!\cdots\!08}{11\!\cdots\!55}a^{6}-\frac{13\!\cdots\!58}{23\!\cdots\!91}a^{5}+\frac{64\!\cdots\!89}{23\!\cdots\!91}a^{4}-\frac{11\!\cdots\!46}{11\!\cdots\!55}a^{3}+\frac{30\!\cdots\!43}{11\!\cdots\!55}a^{2}-\frac{48\!\cdots\!61}{11\!\cdots\!55}a+\frac{58\!\cdots\!19}{182151270203207}$, $\frac{17\!\cdots\!53}{11\!\cdots\!55}a^{23}+\frac{19\!\cdots\!08}{23\!\cdots\!91}a^{22}-\frac{23\!\cdots\!56}{11\!\cdots\!55}a^{21}+\frac{33\!\cdots\!81}{11\!\cdots\!55}a^{20}-\frac{73\!\cdots\!23}{23\!\cdots\!91}a^{19}+\frac{49\!\cdots\!83}{910756351016035}a^{18}-\frac{37\!\cdots\!69}{23\!\cdots\!91}a^{17}+\frac{41\!\cdots\!98}{11\!\cdots\!55}a^{16}-\frac{62\!\cdots\!73}{11\!\cdots\!55}a^{15}+\frac{15\!\cdots\!11}{23\!\cdots\!91}a^{14}-\frac{11\!\cdots\!27}{11\!\cdots\!55}a^{13}+\frac{41\!\cdots\!64}{23\!\cdots\!91}a^{12}-\frac{31\!\cdots\!59}{11\!\cdots\!55}a^{11}+\frac{25\!\cdots\!29}{11\!\cdots\!55}a^{10}+\frac{15\!\cdots\!64}{182151270203207}a^{9}-\frac{62\!\cdots\!39}{11\!\cdots\!55}a^{8}+\frac{19\!\cdots\!73}{23\!\cdots\!91}a^{7}-\frac{95\!\cdots\!53}{11\!\cdots\!55}a^{6}+\frac{65\!\cdots\!68}{11\!\cdots\!55}a^{5}-\frac{65\!\cdots\!08}{23\!\cdots\!91}a^{4}+\frac{11\!\cdots\!28}{11\!\cdots\!55}a^{3}-\frac{60\!\cdots\!06}{23\!\cdots\!91}a^{2}+\frac{48\!\cdots\!16}{11\!\cdots\!55}a-\frac{36\!\cdots\!01}{11\!\cdots\!55}$, $\frac{69\!\cdots\!70}{23\!\cdots\!91}a^{23}-\frac{19\!\cdots\!67}{11\!\cdots\!55}a^{22}+\frac{46\!\cdots\!46}{11\!\cdots\!55}a^{21}-\frac{68\!\cdots\!22}{11\!\cdots\!55}a^{20}+\frac{11\!\cdots\!04}{182151270203207}a^{19}-\frac{26\!\cdots\!57}{23\!\cdots\!91}a^{18}+\frac{38\!\cdots\!51}{11\!\cdots\!55}a^{17}-\frac{84\!\cdots\!83}{11\!\cdots\!55}a^{16}+\frac{12\!\cdots\!96}{11\!\cdots\!55}a^{15}-\frac{31\!\cdots\!35}{23\!\cdots\!91}a^{14}+\frac{45\!\cdots\!35}{23\!\cdots\!91}a^{13}-\frac{42\!\cdots\!73}{11\!\cdots\!55}a^{12}+\frac{64\!\cdots\!69}{11\!\cdots\!55}a^{11}-\frac{40\!\cdots\!31}{910756351016035}a^{10}-\frac{39\!\cdots\!40}{23\!\cdots\!91}a^{9}+\frac{25\!\cdots\!48}{23\!\cdots\!91}a^{8}-\frac{19\!\cdots\!61}{11\!\cdots\!55}a^{7}+\frac{19\!\cdots\!33}{11\!\cdots\!55}a^{6}-\frac{13\!\cdots\!06}{11\!\cdots\!55}a^{5}+\frac{13\!\cdots\!19}{23\!\cdots\!91}a^{4}-\frac{48\!\cdots\!97}{23\!\cdots\!91}a^{3}+\frac{62\!\cdots\!12}{11\!\cdots\!55}a^{2}-\frac{99\!\cdots\!76}{11\!\cdots\!55}a+\frac{70\!\cdots\!42}{11\!\cdots\!55}$, $\frac{38\!\cdots\!61}{23\!\cdots\!91}a^{23}+\frac{98\!\cdots\!23}{11\!\cdots\!55}a^{22}-\frac{16\!\cdots\!73}{910756351016035}a^{21}+\frac{30\!\cdots\!88}{11\!\cdots\!55}a^{20}-\frac{63\!\cdots\!83}{23\!\cdots\!91}a^{19}+\frac{12\!\cdots\!94}{23\!\cdots\!91}a^{18}-\frac{18\!\cdots\!74}{11\!\cdots\!55}a^{17}+\frac{39\!\cdots\!12}{11\!\cdots\!55}a^{16}-\frac{56\!\cdots\!59}{11\!\cdots\!55}a^{15}+\frac{13\!\cdots\!98}{23\!\cdots\!91}a^{14}-\frac{21\!\cdots\!11}{23\!\cdots\!91}a^{13}+\frac{19\!\cdots\!77}{11\!\cdots\!55}a^{12}-\frac{28\!\cdots\!91}{11\!\cdots\!55}a^{11}+\frac{19\!\cdots\!02}{11\!\cdots\!55}a^{10}+\frac{31\!\cdots\!73}{23\!\cdots\!91}a^{9}-\frac{12\!\cdots\!23}{23\!\cdots\!91}a^{8}+\frac{87\!\cdots\!74}{11\!\cdots\!55}a^{7}-\frac{79\!\cdots\!07}{11\!\cdots\!55}a^{6}+\frac{51\!\cdots\!54}{11\!\cdots\!55}a^{5}-\frac{48\!\cdots\!19}{23\!\cdots\!91}a^{4}+\frac{16\!\cdots\!24}{23\!\cdots\!91}a^{3}-\frac{20\!\cdots\!13}{11\!\cdots\!55}a^{2}+\frac{31\!\cdots\!99}{11\!\cdots\!55}a-\frac{18\!\cdots\!66}{910756351016035}$, $\frac{37\!\cdots\!77}{11\!\cdots\!55}a^{23}+\frac{27\!\cdots\!39}{23\!\cdots\!91}a^{22}-\frac{16\!\cdots\!24}{11\!\cdots\!55}a^{21}+\frac{71\!\cdots\!79}{11\!\cdots\!55}a^{20}+\frac{125942035565028}{23\!\cdots\!91}a^{19}+\frac{59\!\cdots\!21}{11\!\cdots\!55}a^{18}-\frac{33\!\cdots\!49}{182151270203207}a^{17}+\frac{28\!\cdots\!87}{11\!\cdots\!55}a^{16}-\frac{18\!\cdots\!32}{11\!\cdots\!55}a^{15}+\frac{27\!\cdots\!34}{23\!\cdots\!91}a^{14}-\frac{64\!\cdots\!18}{11\!\cdots\!55}a^{13}+\frac{29\!\cdots\!89}{23\!\cdots\!91}a^{12}-\frac{88\!\cdots\!51}{11\!\cdots\!55}a^{11}-\frac{22\!\cdots\!29}{11\!\cdots\!55}a^{10}+\frac{11\!\cdots\!72}{23\!\cdots\!91}a^{9}-\frac{60\!\cdots\!31}{11\!\cdots\!55}a^{8}+\frac{41\!\cdots\!68}{23\!\cdots\!91}a^{7}+\frac{25\!\cdots\!88}{11\!\cdots\!55}a^{6}-\frac{33\!\cdots\!16}{910756351016035}a^{5}+\frac{68\!\cdots\!13}{23\!\cdots\!91}a^{4}-\frac{16\!\cdots\!93}{11\!\cdots\!55}a^{3}+\frac{11\!\cdots\!84}{23\!\cdots\!91}a^{2}-\frac{12\!\cdots\!46}{11\!\cdots\!55}a+\frac{12\!\cdots\!91}{11\!\cdots\!55}$, $\frac{64\!\cdots\!42}{11\!\cdots\!55}a^{23}-\frac{45\!\cdots\!19}{11\!\cdots\!55}a^{22}+\frac{13\!\cdots\!06}{11\!\cdots\!55}a^{21}-\frac{22\!\cdots\!43}{11\!\cdots\!55}a^{20}+\frac{50\!\cdots\!25}{23\!\cdots\!91}a^{19}-\frac{27\!\cdots\!52}{910756351016035}a^{18}+\frac{98\!\cdots\!82}{11\!\cdots\!55}a^{17}-\frac{24\!\cdots\!33}{11\!\cdots\!55}a^{16}+\frac{40\!\cdots\!59}{11\!\cdots\!55}a^{15}-\frac{10\!\cdots\!01}{23\!\cdots\!91}a^{14}+\frac{68\!\cdots\!68}{11\!\cdots\!55}a^{13}-\frac{12\!\cdots\!01}{11\!\cdots\!55}a^{12}+\frac{20\!\cdots\!09}{11\!\cdots\!55}a^{11}-\frac{20\!\cdots\!22}{11\!\cdots\!55}a^{10}+\frac{25\!\cdots\!32}{182151270203207}a^{9}+\frac{33\!\cdots\!31}{11\!\cdots\!55}a^{8}-\frac{64\!\cdots\!62}{11\!\cdots\!55}a^{7}+\frac{69\!\cdots\!63}{11\!\cdots\!55}a^{6}-\frac{51\!\cdots\!29}{11\!\cdots\!55}a^{5}+\frac{54\!\cdots\!86}{23\!\cdots\!91}a^{4}-\frac{10\!\cdots\!97}{11\!\cdots\!55}a^{3}+\frac{28\!\cdots\!79}{11\!\cdots\!55}a^{2}-\frac{48\!\cdots\!46}{11\!\cdots\!55}a+\frac{40\!\cdots\!63}{11\!\cdots\!55}$, $\frac{20\!\cdots\!76}{910756351016035}a^{23}+\frac{14\!\cdots\!96}{11\!\cdots\!55}a^{22}-\frac{35\!\cdots\!04}{11\!\cdots\!55}a^{21}+\frac{40\!\cdots\!49}{910756351016035}a^{20}-\frac{11\!\cdots\!82}{23\!\cdots\!91}a^{19}+\frac{99\!\cdots\!94}{11\!\cdots\!55}a^{18}-\frac{28\!\cdots\!53}{11\!\cdots\!55}a^{17}+\frac{63\!\cdots\!67}{11\!\cdots\!55}a^{16}-\frac{96\!\cdots\!46}{11\!\cdots\!55}a^{15}+\frac{23\!\cdots\!16}{23\!\cdots\!91}a^{14}-\frac{17\!\cdots\!07}{11\!\cdots\!55}a^{13}+\frac{32\!\cdots\!49}{11\!\cdots\!55}a^{12}-\frac{48\!\cdots\!51}{11\!\cdots\!55}a^{11}+\frac{40\!\cdots\!38}{11\!\cdots\!55}a^{10}+\frac{27\!\cdots\!84}{23\!\cdots\!91}a^{9}-\frac{95\!\cdots\!94}{11\!\cdots\!55}a^{8}+\frac{15\!\cdots\!23}{11\!\cdots\!55}a^{7}-\frac{15\!\cdots\!07}{11\!\cdots\!55}a^{6}+\frac{10\!\cdots\!01}{11\!\cdots\!55}a^{5}-\frac{10\!\cdots\!84}{23\!\cdots\!91}a^{4}+\frac{19\!\cdots\!28}{11\!\cdots\!55}a^{3}-\frac{48\!\cdots\!11}{11\!\cdots\!55}a^{2}+\frac{77\!\cdots\!54}{11\!\cdots\!55}a-\frac{55\!\cdots\!17}{11\!\cdots\!55}$
| 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: | \( 6022924.686750277 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 6022924.686750277 \cdot 1}{2\cdot\sqrt{641953627807088196277618408203125}}\cr\approx \mathstrut & 0.182366168804516 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 16*x^22 - 26*x^21 + 31*x^20 - 48*x^19 + 128*x^18 - 293*x^17 + 478*x^16 - 623*x^15 + 874*x^14 - 1534*x^13 + 2419*x^12 - 2374*x^11 + 149*x^10 + 3878*x^9 - 7423*x^8 + 8318*x^7 - 6533*x^6 + 3783*x^5 - 1636*x^4 + 521*x^3 - 116*x^2 + 16*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 - 6*x^23 + 16*x^22 - 26*x^21 + 31*x^20 - 48*x^19 + 128*x^18 - 293*x^17 + 478*x^16 - 623*x^15 + 874*x^14 - 1534*x^13 + 2419*x^12 - 2374*x^11 + 149*x^10 + 3878*x^9 - 7423*x^8 + 8318*x^7 - 6533*x^6 + 3783*x^5 - 1636*x^4 + 521*x^3 - 116*x^2 + 16*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 - 6*x^23 + 16*x^22 - 26*x^21 + 31*x^20 - 48*x^19 + 128*x^18 - 293*x^17 + 478*x^16 - 623*x^15 + 874*x^14 - 1534*x^13 + 2419*x^12 - 2374*x^11 + 149*x^10 + 3878*x^9 - 7423*x^8 + 8318*x^7 - 6533*x^6 + 3783*x^5 - 1636*x^4 + 521*x^3 - 116*x^2 + 16*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 - 6*x^23 + 16*x^22 - 26*x^21 + 31*x^20 - 48*x^19 + 128*x^18 - 293*x^17 + 478*x^16 - 623*x^15 + 874*x^14 - 1534*x^13 + 2419*x^12 - 2374*x^11 + 149*x^10 + 3878*x^9 - 7423*x^8 + 8318*x^7 - 6533*x^6 + 3783*x^5 - 1636*x^4 + 521*x^3 - 116*x^2 + 16*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
$\GL(2,5)$ (as 24T1353):
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 480 |
| The 24 conjugacy class representatives for $\GL(2,5)$ |
| Character table for $\GL(2,5)$ |
Intermediate fields
| 6.2.13203125.1, deg 12 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $24$ | $24$ | R | ${\href{/padicField/7.8.0.1}{8} }^{3}$ | $20{,}\,{\href{/padicField/11.4.0.1}{4} }$ | R | ${\href{/padicField/17.8.0.1}{8} }^{3}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | $24$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $24$ | ${\href{/padicField/59.3.0.1}{3} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| Deg $20$ | $5$ | $4$ | $28$ | ||||
|
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |