Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 4*x^22 + x^21 + x^20 - 5*x^19 + 37*x^18 + 33*x^17 - 39*x^16 - 57*x^15 - 111*x^14 - 16*x^13 + 118*x^12 + 13*x^11 + 47*x^9 + 156*x^8 + 319*x^7 + 175*x^6 - 76*x^5 - 127*x^4 - 44*x^3 + 48*x^2 + 56*x + 16)
gp: K = bnfinit(y^24 - y^23 - 4*y^22 + y^21 + y^20 - 5*y^19 + 37*y^18 + 33*y^17 - 39*y^16 - 57*y^15 - 111*y^14 - 16*y^13 + 118*y^12 + 13*y^11 + 47*y^9 + 156*y^8 + 319*y^7 + 175*y^6 - 76*y^5 - 127*y^4 - 44*y^3 + 48*y^2 + 56*y + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - 4*x^22 + x^21 + x^20 - 5*x^19 + 37*x^18 + 33*x^17 - 39*x^16 - 57*x^15 - 111*x^14 - 16*x^13 + 118*x^12 + 13*x^11 + 47*x^9 + 156*x^8 + 319*x^7 + 175*x^6 - 76*x^5 - 127*x^4 - 44*x^3 + 48*x^2 + 56*x + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 - 4*x^22 + x^21 + x^20 - 5*x^19 + 37*x^18 + 33*x^17 - 39*x^16 - 57*x^15 - 111*x^14 - 16*x^13 + 118*x^12 + 13*x^11 + 47*x^9 + 156*x^8 + 319*x^7 + 175*x^6 - 76*x^5 - 127*x^4 - 44*x^3 + 48*x^2 + 56*x + 16)
\( x^{24} - x^{23} - 4 x^{22} + x^{21} + x^{20} - 5 x^{19} + 37 x^{18} + 33 x^{17} - 39 x^{16} - 57 x^{15} + \cdots + 16 \)
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: |
\(34430548576629947662353515625\)
\(\medspace = 3^{12}\cdot 5^{18}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.45\) | 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: | $3^{1/2}5^{3/4}19^{2/3}\approx 41.237329340989255$ | ||
| Ramified primes: |
\(3\), \(5\), \(19\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{1425556}a^{22}+\frac{41357}{1425556}a^{21}-\frac{15043}{712778}a^{20}-\frac{196965}{1425556}a^{19}-\frac{317161}{1425556}a^{18}+\frac{60521}{1425556}a^{17}+\frac{44291}{129596}a^{16}+\frac{682817}{1425556}a^{15}-\frac{353827}{1425556}a^{14}-\frac{55911}{1425556}a^{13}-\frac{519895}{1425556}a^{12}+\frac{135136}{356389}a^{11}+\frac{219189}{712778}a^{10}-\frac{51293}{129596}a^{9}+\frac{324799}{712778}a^{8}-\frac{18529}{1425556}a^{7}+\frac{3869}{32399}a^{6}-\frac{365647}{1425556}a^{5}+\frac{165805}{1425556}a^{4}+\frac{274965}{712778}a^{3}-\frac{405607}{1425556}a^{2}-\frac{29541}{356389}a+\frac{46623}{356389}$, $\frac{1}{85\!\cdots\!28}a^{23}+\frac{18163868979}{85\!\cdots\!28}a^{22}+\frac{10\!\cdots\!72}{10\!\cdots\!41}a^{21}+\frac{15\!\cdots\!13}{85\!\cdots\!28}a^{20}+\frac{12\!\cdots\!29}{85\!\cdots\!28}a^{19}-\frac{878378445445419}{77\!\cdots\!48}a^{18}+\frac{79\!\cdots\!61}{85\!\cdots\!28}a^{17}-\frac{28\!\cdots\!71}{85\!\cdots\!28}a^{16}+\frac{23\!\cdots\!69}{85\!\cdots\!28}a^{15}+\frac{36\!\cdots\!03}{85\!\cdots\!28}a^{14}+\frac{36\!\cdots\!29}{85\!\cdots\!28}a^{13}+\frac{906506742036311}{21\!\cdots\!82}a^{12}-\frac{38\!\cdots\!21}{42\!\cdots\!64}a^{11}-\frac{29\!\cdots\!95}{85\!\cdots\!28}a^{10}+\frac{26\!\cdots\!51}{21\!\cdots\!82}a^{9}+\frac{72\!\cdots\!47}{85\!\cdots\!28}a^{8}+\frac{50\!\cdots\!08}{10\!\cdots\!41}a^{7}+\frac{28\!\cdots\!39}{85\!\cdots\!28}a^{6}-\frac{13\!\cdots\!61}{85\!\cdots\!28}a^{5}+\frac{39\!\cdots\!53}{10\!\cdots\!41}a^{4}+\frac{27\!\cdots\!81}{85\!\cdots\!28}a^{3}+\frac{27\!\cdots\!89}{10\!\cdots\!41}a^{2}-\frac{31\!\cdots\!12}{10\!\cdots\!41}a+\frac{26\!\cdots\!77}{10\!\cdots\!41}$
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: |
\( -\frac{117915814377341653}{42851803183346564} a^{23} + \frac{197207171645330503}{42851803183346564} a^{22} + \frac{85006782973489926}{10712950795836641} a^{21} - \frac{348478004387897971}{42851803183346564} a^{20} + \frac{113679103893968149}{42851803183346564} a^{19} + \frac{518245734125825657}{42851803183346564} a^{18} - \frac{4713653758914993037}{42851803183346564} a^{17} - \frac{727035149700994613}{42851803183346564} a^{16} + \frac{5129588610391762855}{42851803183346564} a^{15} + \frac{3265932441257164717}{42851803183346564} a^{14} + \frac{10829050500302674923}{42851803183346564} a^{13} - \frac{1346668237222617290}{10712950795836641} a^{12} - \frac{2592770900643767350}{10712950795836641} a^{11} + \frac{5523067453432221815}{42851803183346564} a^{10} - \frac{1809883378337436571}{21425901591673282} a^{9} - \frac{17747524028759473}{236750293830644} a^{8} - \frac{8086893335090639107}{21425901591673282} a^{7} - \frac{26743766156315915693}{42851803183346564} a^{6} - \frac{2551875730292042513}{42851803183346564} a^{5} + \frac{2704264347260549967}{10712950795836641} a^{4} + \frac{691362455020807511}{3895618471213324} a^{3} - \frac{25998071151352511}{21425901591673282} a^{2} - \frac{258156184163192927}{1947809235606662} a - \frac{691973979704513672}{10712950795836641} \)
(order $30$)
| 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: |
$\frac{41\!\cdots\!47}{77\!\cdots\!48}a^{23}-\frac{71\!\cdots\!31}{77\!\cdots\!48}a^{22}-\frac{28\!\cdots\!55}{19\!\cdots\!62}a^{21}+\frac{12\!\cdots\!67}{77\!\cdots\!48}a^{20}-\frac{47\!\cdots\!57}{77\!\cdots\!48}a^{19}-\frac{17\!\cdots\!47}{77\!\cdots\!48}a^{18}+\frac{16\!\cdots\!75}{77\!\cdots\!48}a^{17}+\frac{18\!\cdots\!31}{77\!\cdots\!48}a^{16}-\frac{17\!\cdots\!77}{77\!\cdots\!48}a^{15}-\frac{11\!\cdots\!55}{77\!\cdots\!48}a^{14}-\frac{38\!\cdots\!37}{77\!\cdots\!48}a^{13}+\frac{25\!\cdots\!86}{973904617803331}a^{12}+\frac{17\!\cdots\!01}{38\!\cdots\!24}a^{11}-\frac{19\!\cdots\!45}{77\!\cdots\!48}a^{10}+\frac{17\!\cdots\!77}{973904617803331}a^{9}+\frac{96\!\cdots\!01}{77\!\cdots\!48}a^{8}+\frac{14\!\cdots\!17}{19\!\cdots\!62}a^{7}+\frac{91\!\cdots\!29}{77\!\cdots\!48}a^{6}+\frac{73\!\cdots\!17}{77\!\cdots\!48}a^{5}-\frac{91\!\cdots\!37}{19\!\cdots\!62}a^{4}-\frac{26\!\cdots\!21}{77\!\cdots\!48}a^{3}+\frac{11\!\cdots\!31}{19\!\cdots\!62}a^{2}+\frac{24\!\cdots\!40}{973904617803331}a+\frac{11\!\cdots\!65}{973904617803331}$, $\frac{11\!\cdots\!53}{42\!\cdots\!64}a^{23}-\frac{19\!\cdots\!03}{42\!\cdots\!64}a^{22}-\frac{85\!\cdots\!26}{10\!\cdots\!41}a^{21}+\frac{34\!\cdots\!71}{42\!\cdots\!64}a^{20}-\frac{11\!\cdots\!49}{42\!\cdots\!64}a^{19}-\frac{51\!\cdots\!57}{42\!\cdots\!64}a^{18}+\frac{47\!\cdots\!37}{42\!\cdots\!64}a^{17}+\frac{72\!\cdots\!13}{42\!\cdots\!64}a^{16}-\frac{51\!\cdots\!55}{42\!\cdots\!64}a^{15}-\frac{32\!\cdots\!17}{42\!\cdots\!64}a^{14}-\frac{10\!\cdots\!23}{42\!\cdots\!64}a^{13}+\frac{13\!\cdots\!90}{10\!\cdots\!41}a^{12}+\frac{25\!\cdots\!50}{10\!\cdots\!41}a^{11}-\frac{55\!\cdots\!15}{42\!\cdots\!64}a^{10}+\frac{18\!\cdots\!71}{21\!\cdots\!82}a^{9}+\frac{17\!\cdots\!73}{236750293830644}a^{8}+\frac{80\!\cdots\!07}{21\!\cdots\!82}a^{7}+\frac{26\!\cdots\!93}{42\!\cdots\!64}a^{6}+\frac{25\!\cdots\!13}{42\!\cdots\!64}a^{5}-\frac{27\!\cdots\!67}{10\!\cdots\!41}a^{4}-\frac{69\!\cdots\!11}{38\!\cdots\!24}a^{3}+\frac{25\!\cdots\!11}{21\!\cdots\!82}a^{2}+\frac{25\!\cdots\!27}{19\!\cdots\!62}a+\frac{68\!\cdots\!31}{10\!\cdots\!41}$, $\frac{37\!\cdots\!85}{85\!\cdots\!28}a^{23}-\frac{63\!\cdots\!89}{85\!\cdots\!28}a^{22}-\frac{25\!\cdots\!05}{21\!\cdots\!82}a^{21}+\frac{10\!\cdots\!25}{85\!\cdots\!28}a^{20}-\frac{41\!\cdots\!83}{85\!\cdots\!28}a^{19}-\frac{15\!\cdots\!89}{85\!\cdots\!28}a^{18}+\frac{14\!\cdots\!77}{85\!\cdots\!28}a^{17}+\frac{16\!\cdots\!73}{85\!\cdots\!28}a^{16}-\frac{15\!\cdots\!79}{85\!\cdots\!28}a^{15}-\frac{56\!\cdots\!21}{473500587661288}a^{14}-\frac{34\!\cdots\!15}{85\!\cdots\!28}a^{13}+\frac{22\!\cdots\!64}{10\!\cdots\!41}a^{12}+\frac{15\!\cdots\!77}{42\!\cdots\!64}a^{11}-\frac{16\!\cdots\!95}{85\!\cdots\!28}a^{10}+\frac{29\!\cdots\!95}{21\!\cdots\!82}a^{9}+\frac{89\!\cdots\!39}{85\!\cdots\!28}a^{8}+\frac{64\!\cdots\!54}{10\!\cdots\!41}a^{7}+\frac{81\!\cdots\!03}{85\!\cdots\!28}a^{6}+\frac{68\!\cdots\!65}{77\!\cdots\!48}a^{5}-\frac{82\!\cdots\!51}{21\!\cdots\!82}a^{4}-\frac{23\!\cdots\!51}{85\!\cdots\!28}a^{3}+\frac{46\!\cdots\!28}{973904617803331}a^{2}+\frac{43\!\cdots\!89}{21\!\cdots\!82}a+\frac{10\!\cdots\!71}{10\!\cdots\!41}$, $\frac{38\!\cdots\!79}{42\!\cdots\!64}a^{23}-\frac{16\!\cdots\!85}{10\!\cdots\!41}a^{22}-\frac{10\!\cdots\!17}{42\!\cdots\!64}a^{21}+\frac{10\!\cdots\!85}{38\!\cdots\!24}a^{20}-\frac{11\!\cdots\!90}{10\!\cdots\!41}a^{19}-\frac{39\!\cdots\!99}{10\!\cdots\!41}a^{18}+\frac{76\!\cdots\!99}{21\!\cdots\!82}a^{17}+\frac{36\!\cdots\!30}{10\!\cdots\!41}a^{16}-\frac{39\!\cdots\!31}{10\!\cdots\!41}a^{15}-\frac{50\!\cdots\!01}{21\!\cdots\!82}a^{14}-\frac{79\!\cdots\!50}{973904617803331}a^{13}+\frac{19\!\cdots\!39}{42\!\cdots\!64}a^{12}+\frac{15\!\cdots\!17}{21\!\cdots\!82}a^{11}-\frac{17\!\cdots\!43}{42\!\cdots\!64}a^{10}+\frac{12\!\cdots\!91}{42\!\cdots\!64}a^{9}+\frac{85\!\cdots\!51}{42\!\cdots\!64}a^{8}+\frac{53\!\cdots\!27}{42\!\cdots\!64}a^{7}+\frac{82\!\cdots\!15}{42\!\cdots\!64}a^{6}+\frac{31\!\cdots\!35}{21\!\cdots\!82}a^{5}-\frac{33\!\cdots\!49}{42\!\cdots\!64}a^{4}-\frac{23\!\cdots\!85}{42\!\cdots\!64}a^{3}+\frac{68\!\cdots\!91}{42\!\cdots\!64}a^{2}+\frac{44\!\cdots\!48}{10\!\cdots\!41}a+\frac{20\!\cdots\!57}{10\!\cdots\!41}$, $\frac{126579200122743}{42\!\cdots\!64}a^{23}-\frac{580652075312791}{10\!\cdots\!41}a^{22}+\frac{51\!\cdots\!89}{42\!\cdots\!64}a^{21}+\frac{10\!\cdots\!29}{42\!\cdots\!64}a^{20}-\frac{173249387676547}{973904617803331}a^{19}+\frac{54\!\cdots\!39}{21\!\cdots\!82}a^{18}+\frac{816617163798334}{10\!\cdots\!41}a^{17}-\frac{42\!\cdots\!45}{21\!\cdots\!82}a^{16}+\frac{33\!\cdots\!09}{21\!\cdots\!82}a^{15}+\frac{36\!\cdots\!31}{10\!\cdots\!41}a^{14}-\frac{11\!\cdots\!61}{21\!\cdots\!82}a^{13}+\frac{22\!\cdots\!53}{42\!\cdots\!64}a^{12}-\frac{71\!\cdots\!15}{10\!\cdots\!41}a^{11}+\frac{10\!\cdots\!65}{42\!\cdots\!64}a^{10}+\frac{71\!\cdots\!71}{42\!\cdots\!64}a^{9}-\frac{23\!\cdots\!39}{42\!\cdots\!64}a^{8}+\frac{19\!\cdots\!03}{42\!\cdots\!64}a^{7}-\frac{39\!\cdots\!31}{42\!\cdots\!64}a^{6}-\frac{64\!\cdots\!05}{21\!\cdots\!82}a^{5}+\frac{11\!\cdots\!49}{42\!\cdots\!64}a^{4}-\frac{20\!\cdots\!11}{42\!\cdots\!64}a^{3}+\frac{72\!\cdots\!15}{42\!\cdots\!64}a^{2}-\frac{11\!\cdots\!45}{21\!\cdots\!82}a-\frac{73\!\cdots\!91}{10\!\cdots\!41}$, $\frac{43\!\cdots\!05}{42\!\cdots\!64}a^{23}-\frac{76\!\cdots\!57}{42\!\cdots\!64}a^{22}-\frac{59\!\cdots\!61}{21\!\cdots\!82}a^{21}+\frac{13\!\cdots\!31}{42\!\cdots\!64}a^{20}-\frac{55\!\cdots\!55}{42\!\cdots\!64}a^{19}-\frac{17\!\cdots\!43}{42\!\cdots\!64}a^{18}+\frac{17\!\cdots\!63}{42\!\cdots\!64}a^{17}+\frac{13\!\cdots\!75}{42\!\cdots\!64}a^{16}-\frac{18\!\cdots\!61}{42\!\cdots\!64}a^{15}-\frac{11\!\cdots\!63}{42\!\cdots\!64}a^{14}-\frac{40\!\cdots\!97}{42\!\cdots\!64}a^{13}+\frac{11\!\cdots\!85}{21\!\cdots\!82}a^{12}+\frac{86\!\cdots\!22}{10\!\cdots\!41}a^{11}-\frac{11\!\cdots\!43}{236750293830644}a^{10}+\frac{38\!\cdots\!17}{10\!\cdots\!41}a^{9}+\frac{93\!\cdots\!53}{42\!\cdots\!64}a^{8}+\frac{15\!\cdots\!16}{10\!\cdots\!41}a^{7}+\frac{94\!\cdots\!69}{42\!\cdots\!64}a^{6}+\frac{63\!\cdots\!43}{42\!\cdots\!64}a^{5}-\frac{19\!\cdots\!29}{21\!\cdots\!82}a^{4}-\frac{27\!\cdots\!21}{42\!\cdots\!64}a^{3}+\frac{24\!\cdots\!22}{10\!\cdots\!41}a^{2}+\frac{56\!\cdots\!31}{118375146915322}a+\frac{23\!\cdots\!21}{10\!\cdots\!41}$, $\frac{26\!\cdots\!17}{85\!\cdots\!28}a^{23}-\frac{42\!\cdots\!47}{77\!\cdots\!48}a^{22}-\frac{88\!\cdots\!66}{10\!\cdots\!41}a^{21}+\frac{81\!\cdots\!81}{85\!\cdots\!28}a^{20}-\frac{35\!\cdots\!19}{85\!\cdots\!28}a^{19}-\frac{10\!\cdots\!93}{85\!\cdots\!28}a^{18}+\frac{10\!\cdots\!09}{85\!\cdots\!28}a^{17}+\frac{57\!\cdots\!09}{85\!\cdots\!28}a^{16}-\frac{10\!\cdots\!59}{85\!\cdots\!28}a^{15}-\frac{68\!\cdots\!61}{85\!\cdots\!28}a^{14}-\frac{24\!\cdots\!79}{85\!\cdots\!28}a^{13}+\frac{18\!\cdots\!83}{10\!\cdots\!41}a^{12}+\frac{10\!\cdots\!11}{42\!\cdots\!64}a^{11}-\frac{12\!\cdots\!03}{85\!\cdots\!28}a^{10}+\frac{23\!\cdots\!63}{21\!\cdots\!82}a^{9}+\frac{54\!\cdots\!43}{85\!\cdots\!28}a^{8}+\frac{46\!\cdots\!75}{10\!\cdots\!41}a^{7}+\frac{56\!\cdots\!19}{85\!\cdots\!28}a^{6}+\frac{34\!\cdots\!47}{85\!\cdots\!28}a^{5}-\frac{57\!\cdots\!05}{21\!\cdots\!82}a^{4}-\frac{16\!\cdots\!99}{85\!\cdots\!28}a^{3}+\frac{16\!\cdots\!85}{21\!\cdots\!82}a^{2}+\frac{15\!\cdots\!16}{10\!\cdots\!41}a+\frac{69\!\cdots\!58}{10\!\cdots\!41}$, $\frac{96\!\cdots\!83}{10\!\cdots\!41}a^{23}-\frac{16\!\cdots\!65}{10\!\cdots\!41}a^{22}-\frac{53\!\cdots\!09}{21\!\cdots\!82}a^{21}+\frac{57\!\cdots\!55}{21\!\cdots\!82}a^{20}-\frac{11\!\cdots\!28}{10\!\cdots\!41}a^{19}-\frac{80\!\cdots\!07}{21\!\cdots\!82}a^{18}+\frac{77\!\cdots\!45}{21\!\cdots\!82}a^{17}+\frac{76\!\cdots\!15}{21\!\cdots\!82}a^{16}-\frac{80\!\cdots\!31}{21\!\cdots\!82}a^{15}-\frac{51\!\cdots\!95}{21\!\cdots\!82}a^{14}-\frac{17\!\cdots\!11}{21\!\cdots\!82}a^{13}+\frac{97\!\cdots\!49}{21\!\cdots\!82}a^{12}+\frac{15\!\cdots\!03}{21\!\cdots\!82}a^{11}-\frac{44\!\cdots\!32}{10\!\cdots\!41}a^{10}+\frac{32\!\cdots\!06}{10\!\cdots\!41}a^{9}+\frac{44\!\cdots\!17}{21\!\cdots\!82}a^{8}+\frac{13\!\cdots\!09}{10\!\cdots\!41}a^{7}+\frac{42\!\cdots\!33}{21\!\cdots\!82}a^{6}+\frac{16\!\cdots\!67}{10\!\cdots\!41}a^{5}-\frac{17\!\cdots\!37}{21\!\cdots\!82}a^{4}-\frac{11\!\cdots\!59}{19\!\cdots\!62}a^{3}+\frac{18\!\cdots\!91}{10\!\cdots\!41}a^{2}+\frac{82\!\cdots\!21}{19\!\cdots\!62}a+\frac{21\!\cdots\!32}{10\!\cdots\!41}$, $\frac{54\!\cdots\!47}{85\!\cdots\!28}a^{23}-\frac{95\!\cdots\!97}{85\!\cdots\!28}a^{22}-\frac{71\!\cdots\!65}{42\!\cdots\!64}a^{21}+\frac{14\!\cdots\!05}{77\!\cdots\!48}a^{20}-\frac{71\!\cdots\!39}{85\!\cdots\!28}a^{19}-\frac{21\!\cdots\!81}{85\!\cdots\!28}a^{18}+\frac{21\!\cdots\!73}{85\!\cdots\!28}a^{17}+\frac{13\!\cdots\!85}{85\!\cdots\!28}a^{16}-\frac{21\!\cdots\!75}{85\!\cdots\!28}a^{15}-\frac{14\!\cdots\!25}{85\!\cdots\!28}a^{14}-\frac{45\!\cdots\!81}{77\!\cdots\!48}a^{13}+\frac{14\!\cdots\!99}{42\!\cdots\!64}a^{12}+\frac{20\!\cdots\!55}{42\!\cdots\!64}a^{11}-\frac{24\!\cdots\!53}{85\!\cdots\!28}a^{10}+\frac{93\!\cdots\!15}{42\!\cdots\!64}a^{9}+\frac{10\!\cdots\!49}{85\!\cdots\!28}a^{8}+\frac{38\!\cdots\!37}{42\!\cdots\!64}a^{7}+\frac{11\!\cdots\!73}{85\!\cdots\!28}a^{6}+\frac{83\!\cdots\!91}{85\!\cdots\!28}a^{5}-\frac{23\!\cdots\!05}{42\!\cdots\!64}a^{4}-\frac{33\!\cdots\!97}{85\!\cdots\!28}a^{3}+\frac{69\!\cdots\!91}{42\!\cdots\!64}a^{2}+\frac{62\!\cdots\!37}{21\!\cdots\!82}a+\frac{14\!\cdots\!15}{10\!\cdots\!41}$, $\frac{33\!\cdots\!33}{85\!\cdots\!28}a^{23}-\frac{57\!\cdots\!47}{85\!\cdots\!28}a^{22}-\frac{46\!\cdots\!69}{42\!\cdots\!64}a^{21}+\frac{10\!\cdots\!01}{85\!\cdots\!28}a^{20}-\frac{37\!\cdots\!61}{85\!\cdots\!28}a^{19}-\frac{14\!\cdots\!03}{85\!\cdots\!28}a^{18}+\frac{13\!\cdots\!67}{85\!\cdots\!28}a^{17}+\frac{15\!\cdots\!07}{85\!\cdots\!28}a^{16}-\frac{14\!\cdots\!01}{85\!\cdots\!28}a^{15}-\frac{89\!\cdots\!27}{85\!\cdots\!28}a^{14}-\frac{30\!\cdots\!69}{85\!\cdots\!28}a^{13}+\frac{82\!\cdots\!65}{42\!\cdots\!64}a^{12}+\frac{14\!\cdots\!55}{42\!\cdots\!64}a^{11}-\frac{15\!\cdots\!87}{85\!\cdots\!28}a^{10}+\frac{54\!\cdots\!15}{42\!\cdots\!64}a^{9}+\frac{81\!\cdots\!59}{85\!\cdots\!28}a^{8}+\frac{23\!\cdots\!77}{42\!\cdots\!64}a^{7}+\frac{74\!\cdots\!67}{85\!\cdots\!28}a^{6}+\frac{58\!\cdots\!17}{85\!\cdots\!28}a^{5}-\frac{15\!\cdots\!85}{42\!\cdots\!64}a^{4}-\frac{19\!\cdots\!57}{77\!\cdots\!48}a^{3}+\frac{26\!\cdots\!87}{42\!\cdots\!64}a^{2}+\frac{17\!\cdots\!80}{973904617803331}a+\frac{93\!\cdots\!01}{10\!\cdots\!41}$, $\frac{11\!\cdots\!51}{38\!\cdots\!24}a^{23}-\frac{11\!\cdots\!23}{21\!\cdots\!82}a^{22}-\frac{36\!\cdots\!59}{42\!\cdots\!64}a^{21}+\frac{39\!\cdots\!39}{42\!\cdots\!64}a^{20}-\frac{37\!\cdots\!03}{10\!\cdots\!41}a^{19}-\frac{27\!\cdots\!11}{21\!\cdots\!82}a^{18}+\frac{26\!\cdots\!09}{21\!\cdots\!82}a^{17}+\frac{12\!\cdots\!51}{973904617803331}a^{16}-\frac{27\!\cdots\!69}{21\!\cdots\!82}a^{15}-\frac{17\!\cdots\!13}{21\!\cdots\!82}a^{14}-\frac{60\!\cdots\!93}{21\!\cdots\!82}a^{13}+\frac{65\!\cdots\!77}{42\!\cdots\!64}a^{12}+\frac{26\!\cdots\!14}{10\!\cdots\!41}a^{11}-\frac{60\!\cdots\!41}{42\!\cdots\!64}a^{10}+\frac{39\!\cdots\!57}{38\!\cdots\!24}a^{9}+\frac{30\!\cdots\!55}{42\!\cdots\!64}a^{8}+\frac{18\!\cdots\!39}{42\!\cdots\!64}a^{7}+\frac{25\!\cdots\!35}{38\!\cdots\!24}a^{6}+\frac{57\!\cdots\!61}{10\!\cdots\!41}a^{5}-\frac{11\!\cdots\!25}{42\!\cdots\!64}a^{4}-\frac{82\!\cdots\!03}{42\!\cdots\!64}a^{3}+\frac{19\!\cdots\!19}{42\!\cdots\!64}a^{2}+\frac{30\!\cdots\!39}{21\!\cdots\!82}a+\frac{72\!\cdots\!67}{10\!\cdots\!41}$
| 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: | \( 242614.53860678803 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 242614.53860678803 \cdot 1}{30\cdot\sqrt{34430548576629947662353515625}}\cr\approx \mathstrut & 0.164999267690649 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 4*x^22 + x^21 + x^20 - 5*x^19 + 37*x^18 + 33*x^17 - 39*x^16 - 57*x^15 - 111*x^14 - 16*x^13 + 118*x^12 + 13*x^11 + 47*x^9 + 156*x^8 + 319*x^7 + 175*x^6 - 76*x^5 - 127*x^4 - 44*x^3 + 48*x^2 + 56*x + 16)
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 - 4*x^22 + x^21 + x^20 - 5*x^19 + 37*x^18 + 33*x^17 - 39*x^16 - 57*x^15 - 111*x^14 - 16*x^13 + 118*x^12 + 13*x^11 + 47*x^9 + 156*x^8 + 319*x^7 + 175*x^6 - 76*x^5 - 127*x^4 - 44*x^3 + 48*x^2 + 56*x + 16, 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 - 4*x^22 + x^21 + x^20 - 5*x^19 + 37*x^18 + 33*x^17 - 39*x^16 - 57*x^15 - 111*x^14 - 16*x^13 + 118*x^12 + 13*x^11 + 47*x^9 + 156*x^8 + 319*x^7 + 175*x^6 - 76*x^5 - 127*x^4 - 44*x^3 + 48*x^2 + 56*x + 16);
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 - 4*x^22 + x^21 + x^20 - 5*x^19 + 37*x^18 + 33*x^17 - 39*x^16 - 57*x^15 - 111*x^14 - 16*x^13 + 118*x^12 + 13*x^11 + 47*x^9 + 156*x^8 + 319*x^7 + 175*x^6 - 76*x^5 - 127*x^4 - 44*x^3 + 48*x^2 + 56*x + 16);
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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.1218375.1, \(\Q(\zeta_{15})\), 12.0.1484437640625.1 |
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }^{3}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{12}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $24$ | $2$ | $12$ | $12$ | |||
|
\(5\)
| Deg $24$ | $4$ | $6$ | $18$ | |||
|
\(19\)
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 19.6.4.1 | $x^{6} + 304 x^{3} - 5415$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 19.6.4.1 | $x^{6} + 304 x^{3} - 5415$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.285.6t1.a.a | $1$ | $ 3 \cdot 5 \cdot 19 $ | 6.0.439833375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.95.6t1.a.a | $1$ | $ 5 \cdot 19 $ | 6.6.16290125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.57.6t1.a.a | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.95.6t1.a.b | $1$ | $ 5 \cdot 19 $ | 6.6.16290125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.285.6t1.a.b | $1$ | $ 3 \cdot 5 \cdot 19 $ | 6.0.439833375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.57.6t1.a.b | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (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) | $0$ | $-1$ |
| * | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.95.12t1.a.a | $1$ | $ 5 \cdot 19 $ | 12.0.33171021564453125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.95.12t1.a.b | $1$ | $ 5 \cdot 19 $ | 12.0.33171021564453125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.95.12t1.a.c | $1$ | $ 5 \cdot 19 $ | 12.0.33171021564453125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.95.12t1.a.d | $1$ | $ 5 \cdot 19 $ | 12.0.33171021564453125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.285.12t1.a.a | $1$ | $ 3 \cdot 5 \cdot 19 $ | 12.12.24181674720486328125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.285.12t1.a.b | $1$ | $ 3 \cdot 5 \cdot 19 $ | 12.12.24181674720486328125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.285.12t1.a.c | $1$ | $ 3 \cdot 5 \cdot 19 $ | 12.12.24181674720486328125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.285.12t1.a.d | $1$ | $ 3 \cdot 5 \cdot 19 $ | 12.12.24181674720486328125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 2.5415.3t2.a.a | $2$ | $ 3 \cdot 5 \cdot 19^{2}$ | 3.1.5415.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.5415.6t3.d.a | $2$ | $ 3 \cdot 5 \cdot 19^{2}$ | 6.2.146611125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.285.6t5.c.a | $2$ | $ 3 \cdot 5 \cdot 19 $ | 6.0.1218375.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.285.12t18.a.a | $2$ | $ 3 \cdot 5 \cdot 19 $ | 12.0.1484437640625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.285.12t18.a.b | $2$ | $ 3 \cdot 5 \cdot 19 $ | 12.0.1484437640625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.285.6t5.c.b | $2$ | $ 3 \cdot 5 \cdot 19 $ | 6.0.1218375.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| 2.27075.12t11.b.a | $2$ | $ 3 \cdot 5^{2} \cdot 19^{2}$ | 12.4.24181674720486328125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.27075.12t11.b.b | $2$ | $ 3 \cdot 5^{2} \cdot 19^{2}$ | 12.4.24181674720486328125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.1425.24t65.a.a | $2$ | $ 3 \cdot 5^{2} \cdot 19 $ | 24.0.34430548576629947662353515625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.1425.24t65.a.b | $2$ | $ 3 \cdot 5^{2} \cdot 19 $ | 24.0.34430548576629947662353515625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.1425.24t65.a.c | $2$ | $ 3 \cdot 5^{2} \cdot 19 $ | 24.0.34430548576629947662353515625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.1425.24t65.a.d | $2$ | $ 3 \cdot 5^{2} \cdot 19 $ | 24.0.34430548576629947662353515625.1 | $S_3\times C_{12}$ (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 *.