Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 11*x^22 + 47*x^21 - 78*x^20 - 51*x^19 + 367*x^18 - 359*x^17 - 364*x^16 + 1182*x^15 - 712*x^14 - 1586*x^13 + 3310*x^12 - 393*x^11 - 5271*x^10 + 5059*x^9 + 3246*x^8 - 8091*x^7 - 125*x^6 + 10047*x^5 - 6818*x^4 - 2608*x^3 + 2872*x^2 + 1488*x - 2176)
gp: K = bnfinit(y^23 - 11*y^22 + 47*y^21 - 78*y^20 - 51*y^19 + 367*y^18 - 359*y^17 - 364*y^16 + 1182*y^15 - 712*y^14 - 1586*y^13 + 3310*y^12 - 393*y^11 - 5271*y^10 + 5059*y^9 + 3246*y^8 - 8091*y^7 - 125*y^6 + 10047*y^5 - 6818*y^4 - 2608*y^3 + 2872*y^2 + 1488*y - 2176, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 11*x^22 + 47*x^21 - 78*x^20 - 51*x^19 + 367*x^18 - 359*x^17 - 364*x^16 + 1182*x^15 - 712*x^14 - 1586*x^13 + 3310*x^12 - 393*x^11 - 5271*x^10 + 5059*x^9 + 3246*x^8 - 8091*x^7 - 125*x^6 + 10047*x^5 - 6818*x^4 - 2608*x^3 + 2872*x^2 + 1488*x - 2176);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 11*x^22 + 47*x^21 - 78*x^20 - 51*x^19 + 367*x^18 - 359*x^17 - 364*x^16 + 1182*x^15 - 712*x^14 - 1586*x^13 + 3310*x^12 - 393*x^11 - 5271*x^10 + 5059*x^9 + 3246*x^8 - 8091*x^7 - 125*x^6 + 10047*x^5 - 6818*x^4 - 2608*x^3 + 2872*x^2 + 1488*x - 2176)
\( x^{23} - 11 x^{22} + 47 x^{21} - 78 x^{20} - 51 x^{19} + 367 x^{18} - 359 x^{17} - 364 x^{16} + \cdots - 2176 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1823491470737247969166873376199879779\)
\(\medspace = -\,1979^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(37.72\) | 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: | $1979^{1/2}\approx 44.48595283907045$ | ||
| Ramified primes: |
\(1979\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1979}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{5}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{9}-\frac{1}{8}a^{6}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{13}-\frac{1}{8}a^{11}-\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{3}{16}a^{7}+\frac{1}{8}a^{5}-\frac{3}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{16}a^{8}-\frac{3}{16}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{16}a^{9}-\frac{3}{16}a^{6}+\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{16}+\frac{1}{16}a^{10}-\frac{1}{4}a^{7}-\frac{3}{32}a^{4}+\frac{1}{4}a$, $\frac{1}{32}a^{17}+\frac{1}{16}a^{11}-\frac{3}{32}a^{5}$, $\frac{1}{32}a^{18}-\frac{1}{16}a^{12}-\frac{1}{8}a^{9}+\frac{1}{32}a^{6}+\frac{1}{8}a^{3}$, $\frac{1}{32}a^{19}-\frac{1}{8}a^{11}+\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{7}{32}a^{7}+\frac{1}{8}a^{5}+\frac{3}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{20}-\frac{1}{64}a^{17}-\frac{1}{32}a^{14}-\frac{3}{32}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{7}{64}a^{8}-\frac{9}{64}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{2542991104}a^{21}+\frac{15711945}{2542991104}a^{20}-\frac{85999}{5941568}a^{19}+\frac{38623339}{2542991104}a^{18}-\frac{20830419}{2542991104}a^{17}-\frac{2662609}{1271495552}a^{16}-\frac{2711505}{1271495552}a^{15}-\frac{7851203}{1271495552}a^{14}+\frac{9104205}{635747776}a^{13}-\frac{10476871}{1271495552}a^{12}+\frac{127819749}{1271495552}a^{11}+\frac{31162009}{317873888}a^{10}+\frac{225510353}{2542991104}a^{9}-\frac{81803051}{2542991104}a^{8}-\frac{37056}{584327}a^{7}+\frac{53996883}{2542991104}a^{6}+\frac{140776233}{2542991104}a^{5}+\frac{230558005}{1271495552}a^{4}-\frac{15188691}{39734236}a^{3}+\frac{67112569}{317873888}a^{2}+\frac{5690731}{158936944}a+\frac{399831}{1168654}$, $\frac{1}{72\!\cdots\!08}a^{22}-\frac{32236769}{36\!\cdots\!04}a^{21}-\frac{12\!\cdots\!55}{72\!\cdots\!08}a^{20}-\frac{36\!\cdots\!13}{72\!\cdots\!08}a^{19}+\frac{595055344691565}{10\!\cdots\!56}a^{18}-\frac{68\!\cdots\!53}{72\!\cdots\!08}a^{17}+\frac{10\!\cdots\!97}{18\!\cdots\!52}a^{16}-\frac{17\!\cdots\!65}{56\!\cdots\!36}a^{15}-\frac{22\!\cdots\!81}{97\!\cdots\!92}a^{14}+\frac{91\!\cdots\!47}{36\!\cdots\!04}a^{13}+\frac{81\!\cdots\!01}{18\!\cdots\!52}a^{12}-\frac{28\!\cdots\!47}{36\!\cdots\!04}a^{11}+\frac{80\!\cdots\!29}{72\!\cdots\!08}a^{10}+\frac{14\!\cdots\!65}{36\!\cdots\!04}a^{9}-\frac{39\!\cdots\!03}{72\!\cdots\!08}a^{8}+\frac{18\!\cdots\!03}{72\!\cdots\!08}a^{7}-\frac{75\!\cdots\!15}{90\!\cdots\!76}a^{6}+\frac{13\!\cdots\!31}{72\!\cdots\!08}a^{5}+\frac{84\!\cdots\!41}{36\!\cdots\!04}a^{4}-\frac{72\!\cdots\!67}{90\!\cdots\!76}a^{3}-\frac{10\!\cdots\!69}{90\!\cdots\!76}a^{2}+\frac{88\!\cdots\!59}{45\!\cdots\!88}a+\frac{628475915758957}{33\!\cdots\!08}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{27\!\cdots\!81}{72\!\cdots\!08}a^{22}-\frac{12\!\cdots\!09}{36\!\cdots\!04}a^{21}+\frac{67\!\cdots\!53}{72\!\cdots\!08}a^{20}-\frac{14\!\cdots\!05}{72\!\cdots\!08}a^{19}-\frac{79\!\cdots\!75}{18\!\cdots\!52}a^{18}+\frac{34\!\cdots\!71}{72\!\cdots\!08}a^{17}+\frac{11\!\cdots\!57}{18\!\cdots\!52}a^{16}-\frac{66\!\cdots\!57}{52\!\cdots\!08}a^{15}+\frac{72\!\cdots\!47}{97\!\cdots\!92}a^{14}+\frac{74\!\cdots\!39}{36\!\cdots\!04}a^{13}-\frac{71\!\cdots\!51}{18\!\cdots\!52}a^{12}-\frac{34\!\cdots\!99}{36\!\cdots\!04}a^{11}+\frac{55\!\cdots\!89}{72\!\cdots\!08}a^{10}-\frac{11\!\cdots\!55}{21\!\cdots\!12}a^{9}-\frac{58\!\cdots\!83}{72\!\cdots\!08}a^{8}+\frac{75\!\cdots\!71}{72\!\cdots\!08}a^{7}+\frac{49\!\cdots\!53}{90\!\cdots\!76}a^{6}-\frac{10\!\cdots\!65}{72\!\cdots\!08}a^{5}-\frac{54\!\cdots\!55}{36\!\cdots\!04}a^{4}+\frac{47\!\cdots\!33}{90\!\cdots\!76}a^{3}-\frac{15\!\cdots\!05}{53\!\cdots\!28}a^{2}-\frac{26\!\cdots\!77}{45\!\cdots\!88}a-\frac{39\!\cdots\!11}{33\!\cdots\!08}$, $\frac{1312877}{198733566464}a^{22}+\frac{489703}{1552605988}a^{21}-\frac{556376337}{198733566464}a^{20}+\frac{1398290779}{198733566464}a^{19}+\frac{341384881}{99366783232}a^{18}-\frac{53126573}{1564831232}a^{17}+\frac{48933935}{6210423952}a^{16}+\frac{3879318971}{49683391616}a^{15}-\frac{55020363}{2685588736}a^{14}-\frac{86335301}{5845104896}a^{13}+\frac{3867915563}{24841695808}a^{12}-\frac{19128230773}{99366783232}a^{11}-\frac{60419370379}{198733566464}a^{10}+\frac{23461766087}{49683391616}a^{9}+\frac{64122051623}{198733566464}a^{8}-\frac{135048553057}{198733566464}a^{7}-\frac{2982464549}{99366783232}a^{6}+\frac{186121124909}{198733566464}a^{5}-\frac{52139632561}{99366783232}a^{4}-\frac{19063356371}{24841695808}a^{3}+\frac{6463642049}{24841695808}a^{2}+\frac{12314182017}{12420847904}a-\frac{51093361}{91329764}$, $\frac{26\!\cdots\!85}{72\!\cdots\!08}a^{22}-\frac{82714895763453}{21\!\cdots\!92}a^{21}+\frac{11\!\cdots\!83}{72\!\cdots\!08}a^{20}-\frac{14\!\cdots\!49}{72\!\cdots\!08}a^{19}-\frac{13\!\cdots\!51}{36\!\cdots\!04}a^{18}+\frac{93\!\cdots\!05}{72\!\cdots\!08}a^{17}-\frac{20\!\cdots\!35}{45\!\cdots\!88}a^{16}-\frac{41\!\cdots\!09}{18\!\cdots\!52}a^{15}+\frac{31\!\cdots\!97}{97\!\cdots\!92}a^{14}+\frac{10\!\cdots\!11}{36\!\cdots\!04}a^{13}-\frac{64\!\cdots\!77}{90\!\cdots\!76}a^{12}+\frac{28\!\cdots\!07}{36\!\cdots\!04}a^{11}+\frac{45\!\cdots\!73}{72\!\cdots\!08}a^{10}-\frac{35\!\cdots\!93}{18\!\cdots\!52}a^{9}+\frac{27\!\cdots\!87}{72\!\cdots\!08}a^{8}+\frac{17\!\cdots\!83}{72\!\cdots\!08}a^{7}-\frac{53\!\cdots\!57}{36\!\cdots\!04}a^{6}-\frac{16\!\cdots\!71}{72\!\cdots\!08}a^{5}+\frac{97\!\cdots\!23}{36\!\cdots\!04}a^{4}+\frac{39\!\cdots\!97}{90\!\cdots\!76}a^{3}-\frac{11\!\cdots\!27}{90\!\cdots\!76}a^{2}-\frac{76\!\cdots\!87}{45\!\cdots\!88}a+\frac{26\!\cdots\!83}{33\!\cdots\!08}$, $\frac{13\!\cdots\!25}{72\!\cdots\!08}a^{22}-\frac{74\!\cdots\!03}{36\!\cdots\!04}a^{21}+\frac{64\!\cdots\!97}{72\!\cdots\!08}a^{20}-\frac{96\!\cdots\!73}{72\!\cdots\!08}a^{19}-\frac{78\!\cdots\!93}{45\!\cdots\!88}a^{18}+\frac{56\!\cdots\!63}{72\!\cdots\!08}a^{17}-\frac{76\!\cdots\!45}{18\!\cdots\!52}a^{16}-\frac{11\!\cdots\!53}{90\!\cdots\!76}a^{15}+\frac{20\!\cdots\!31}{97\!\cdots\!92}a^{14}-\frac{17\!\cdots\!61}{36\!\cdots\!04}a^{13}-\frac{77\!\cdots\!97}{18\!\cdots\!52}a^{12}+\frac{20\!\cdots\!13}{36\!\cdots\!04}a^{11}+\frac{19\!\cdots\!99}{67\!\cdots\!44}a^{10}-\frac{43\!\cdots\!41}{36\!\cdots\!04}a^{9}+\frac{35\!\cdots\!05}{72\!\cdots\!08}a^{8}+\frac{91\!\cdots\!75}{72\!\cdots\!08}a^{7}-\frac{24\!\cdots\!31}{18\!\cdots\!52}a^{6}-\frac{70\!\cdots\!41}{72\!\cdots\!08}a^{5}+\frac{78\!\cdots\!69}{36\!\cdots\!04}a^{4}-\frac{28\!\cdots\!71}{53\!\cdots\!28}a^{3}-\frac{90\!\cdots\!61}{90\!\cdots\!76}a^{2}+\frac{98\!\cdots\!27}{45\!\cdots\!88}a+\frac{23\!\cdots\!09}{33\!\cdots\!08}$, $\frac{21\!\cdots\!57}{36\!\cdots\!04}a^{22}-\frac{10\!\cdots\!31}{18\!\cdots\!52}a^{21}+\frac{69\!\cdots\!09}{36\!\cdots\!04}a^{20}-\frac{53\!\cdots\!25}{36\!\cdots\!04}a^{19}-\frac{27\!\cdots\!53}{45\!\cdots\!88}a^{18}+\frac{47\!\cdots\!71}{36\!\cdots\!04}a^{17}+\frac{14\!\cdots\!51}{90\!\cdots\!76}a^{16}-\frac{11\!\cdots\!63}{45\!\cdots\!88}a^{15}+\frac{14\!\cdots\!43}{48\!\cdots\!96}a^{14}+\frac{22\!\cdots\!35}{18\!\cdots\!52}a^{13}-\frac{77\!\cdots\!09}{90\!\cdots\!76}a^{12}+\frac{11\!\cdots\!97}{18\!\cdots\!52}a^{11}+\frac{36\!\cdots\!37}{36\!\cdots\!04}a^{10}-\frac{32\!\cdots\!73}{18\!\cdots\!52}a^{9}-\frac{42\!\cdots\!55}{36\!\cdots\!04}a^{8}+\frac{86\!\cdots\!11}{36\!\cdots\!04}a^{7}-\frac{10\!\cdots\!45}{90\!\cdots\!76}a^{6}-\frac{55\!\cdots\!45}{21\!\cdots\!12}a^{5}+\frac{42\!\cdots\!29}{18\!\cdots\!52}a^{4}+\frac{24\!\cdots\!21}{45\!\cdots\!88}a^{3}-\frac{56\!\cdots\!41}{45\!\cdots\!88}a^{2}-\frac{94\!\cdots\!69}{22\!\cdots\!44}a+\frac{11\!\cdots\!67}{16\!\cdots\!54}$, $\frac{15\!\cdots\!85}{36\!\cdots\!04}a^{22}-\frac{19\!\cdots\!21}{45\!\cdots\!88}a^{21}+\frac{32\!\cdots\!75}{21\!\cdots\!12}a^{20}-\frac{50\!\cdots\!17}{36\!\cdots\!04}a^{19}-\frac{84\!\cdots\!35}{18\!\cdots\!52}a^{18}+\frac{96\!\cdots\!67}{83\!\cdots\!28}a^{17}-\frac{169261098236713}{56\!\cdots\!36}a^{16}-\frac{20\!\cdots\!49}{90\!\cdots\!76}a^{15}+\frac{14\!\cdots\!49}{48\!\cdots\!96}a^{14}+\frac{17\!\cdots\!59}{18\!\cdots\!52}a^{13}-\frac{35\!\cdots\!53}{45\!\cdots\!88}a^{12}+\frac{11\!\cdots\!31}{18\!\cdots\!52}a^{11}+\frac{16\!\cdots\!33}{21\!\cdots\!12}a^{10}-\frac{15\!\cdots\!75}{90\!\cdots\!76}a^{9}+\frac{45\!\cdots\!39}{36\!\cdots\!04}a^{8}+\frac{79\!\cdots\!91}{36\!\cdots\!04}a^{7}-\frac{24\!\cdots\!41}{18\!\cdots\!52}a^{6}-\frac{80\!\cdots\!23}{36\!\cdots\!04}a^{5}+\frac{47\!\cdots\!43}{18\!\cdots\!52}a^{4}+\frac{23\!\cdots\!35}{10\!\cdots\!16}a^{3}-\frac{55\!\cdots\!87}{45\!\cdots\!88}a^{2}-\frac{35\!\cdots\!31}{22\!\cdots\!44}a+\frac{13\!\cdots\!41}{16\!\cdots\!54}$, $\frac{61\!\cdots\!61}{72\!\cdots\!08}a^{22}-\frac{35\!\cdots\!89}{36\!\cdots\!04}a^{21}+\frac{30\!\cdots\!97}{72\!\cdots\!08}a^{20}-\frac{51\!\cdots\!97}{72\!\cdots\!08}a^{19}-\frac{10\!\cdots\!37}{18\!\cdots\!52}a^{18}+\frac{15\!\cdots\!95}{42\!\cdots\!24}a^{17}-\frac{54\!\cdots\!03}{18\!\cdots\!52}a^{16}-\frac{23\!\cdots\!85}{45\!\cdots\!88}a^{15}+\frac{11\!\cdots\!83}{97\!\cdots\!92}a^{14}-\frac{13\!\cdots\!61}{36\!\cdots\!04}a^{13}-\frac{19\!\cdots\!35}{10\!\cdots\!56}a^{12}+\frac{11\!\cdots\!25}{36\!\cdots\!04}a^{11}+\frac{37\!\cdots\!09}{72\!\cdots\!08}a^{10}-\frac{21\!\cdots\!15}{36\!\cdots\!04}a^{9}+\frac{25\!\cdots\!29}{72\!\cdots\!08}a^{8}+\frac{40\!\cdots\!11}{72\!\cdots\!08}a^{7}-\frac{19\!\cdots\!05}{26\!\cdots\!64}a^{6}-\frac{23\!\cdots\!29}{72\!\cdots\!08}a^{5}+\frac{40\!\cdots\!25}{36\!\cdots\!04}a^{4}-\frac{23\!\cdots\!59}{90\!\cdots\!76}a^{3}-\frac{54\!\cdots\!61}{90\!\cdots\!76}a^{2}+\frac{16\!\cdots\!39}{45\!\cdots\!88}a+\frac{10\!\cdots\!61}{33\!\cdots\!08}$, $\frac{15\!\cdots\!13}{36\!\cdots\!04}a^{22}-\frac{16\!\cdots\!59}{36\!\cdots\!04}a^{21}+\frac{45\!\cdots\!71}{22\!\cdots\!44}a^{20}-\frac{14\!\cdots\!57}{36\!\cdots\!04}a^{19}+\frac{27\!\cdots\!57}{36\!\cdots\!04}a^{18}+\frac{19\!\cdots\!57}{18\!\cdots\!52}a^{17}-\frac{35\!\cdots\!17}{18\!\cdots\!52}a^{16}+\frac{13\!\cdots\!25}{18\!\cdots\!52}a^{15}+\frac{48\!\cdots\!95}{14\!\cdots\!44}a^{14}-\frac{98\!\cdots\!19}{18\!\cdots\!52}a^{13}-\frac{36\!\cdots\!75}{18\!\cdots\!52}a^{12}+\frac{23\!\cdots\!99}{22\!\cdots\!44}a^{11}-\frac{43\!\cdots\!15}{36\!\cdots\!04}a^{10}-\frac{15\!\cdots\!95}{36\!\cdots\!04}a^{9}+\frac{20\!\cdots\!65}{90\!\cdots\!76}a^{8}-\frac{58\!\cdots\!09}{36\!\cdots\!04}a^{7}-\frac{39\!\cdots\!23}{36\!\cdots\!04}a^{6}+\frac{36\!\cdots\!99}{18\!\cdots\!52}a^{5}+\frac{22\!\cdots\!61}{66\!\cdots\!16}a^{4}-\frac{16\!\cdots\!27}{45\!\cdots\!88}a^{3}+\frac{79\!\cdots\!81}{22\!\cdots\!44}a^{2}-\frac{22\!\cdots\!27}{14\!\cdots\!59}a-\frac{11\!\cdots\!01}{829962586718827}$, $\frac{20\!\cdots\!61}{72\!\cdots\!08}a^{22}-\frac{10\!\cdots\!11}{36\!\cdots\!04}a^{21}+\frac{89\!\cdots\!85}{72\!\cdots\!08}a^{20}-\frac{13\!\cdots\!37}{72\!\cdots\!08}a^{19}-\frac{33\!\cdots\!99}{22\!\cdots\!44}a^{18}+\frac{60\!\cdots\!71}{72\!\cdots\!08}a^{17}-\frac{12\!\cdots\!69}{18\!\cdots\!52}a^{16}-\frac{56\!\cdots\!01}{90\!\cdots\!76}a^{15}+\frac{11\!\cdots\!63}{57\!\cdots\!76}a^{14}-\frac{48\!\cdots\!61}{36\!\cdots\!04}a^{13}-\frac{46\!\cdots\!21}{18\!\cdots\!52}a^{12}+\frac{17\!\cdots\!89}{36\!\cdots\!04}a^{11}+\frac{36\!\cdots\!21}{72\!\cdots\!08}a^{10}-\frac{26\!\cdots\!29}{36\!\cdots\!04}a^{9}+\frac{28\!\cdots\!85}{72\!\cdots\!08}a^{8}+\frac{49\!\cdots\!35}{72\!\cdots\!08}a^{7}-\frac{10\!\cdots\!87}{18\!\cdots\!52}a^{6}-\frac{78\!\cdots\!21}{72\!\cdots\!08}a^{5}+\frac{34\!\cdots\!13}{21\!\cdots\!12}a^{4}-\frac{93\!\cdots\!87}{90\!\cdots\!76}a^{3}-\frac{52\!\cdots\!97}{90\!\cdots\!76}a^{2}+\frac{17\!\cdots\!35}{45\!\cdots\!88}a+\frac{17\!\cdots\!89}{33\!\cdots\!08}$, $\frac{36416063585547}{11\!\cdots\!52}a^{22}-\frac{34\!\cdots\!17}{97\!\cdots\!92}a^{21}+\frac{28\!\cdots\!23}{19\!\cdots\!84}a^{20}-\frac{39\!\cdots\!79}{19\!\cdots\!84}a^{19}-\frac{909863371419515}{30\!\cdots\!56}a^{18}+\frac{23\!\cdots\!93}{19\!\cdots\!84}a^{17}-\frac{30\!\cdots\!47}{48\!\cdots\!96}a^{16}-\frac{47\!\cdots\!31}{24\!\cdots\!48}a^{15}+\frac{31\!\cdots\!61}{97\!\cdots\!92}a^{14}-\frac{60\!\cdots\!19}{97\!\cdots\!92}a^{13}-\frac{32\!\cdots\!79}{48\!\cdots\!96}a^{12}+\frac{84\!\cdots\!63}{97\!\cdots\!92}a^{11}+\frac{86\!\cdots\!87}{19\!\cdots\!84}a^{10}-\frac{18\!\cdots\!23}{97\!\cdots\!92}a^{9}+\frac{14\!\cdots\!87}{19\!\cdots\!84}a^{8}+\frac{22\!\cdots\!81}{11\!\cdots\!52}a^{7}-\frac{98\!\cdots\!29}{48\!\cdots\!96}a^{6}-\frac{29\!\cdots\!19}{19\!\cdots\!84}a^{5}+\frac{31\!\cdots\!43}{97\!\cdots\!92}a^{4}-\frac{49\!\cdots\!61}{24\!\cdots\!48}a^{3}-\frac{35\!\cdots\!91}{24\!\cdots\!48}a^{2}+\frac{35\!\cdots\!61}{12\!\cdots\!24}a+\frac{919975643842575}{89725685050684}$, $\frac{55\!\cdots\!91}{36\!\cdots\!04}a^{22}-\frac{13\!\cdots\!35}{90\!\cdots\!76}a^{21}+\frac{17\!\cdots\!57}{36\!\cdots\!04}a^{20}-\frac{14\!\cdots\!27}{36\!\cdots\!04}a^{19}-\frac{26\!\cdots\!91}{18\!\cdots\!52}a^{18}+\frac{11\!\cdots\!23}{36\!\cdots\!04}a^{17}-\frac{11\!\cdots\!07}{45\!\cdots\!88}a^{16}-\frac{52\!\cdots\!05}{90\!\cdots\!76}a^{15}+\frac{41\!\cdots\!59}{48\!\cdots\!96}a^{14}+\frac{52\!\cdots\!33}{18\!\cdots\!52}a^{13}-\frac{35\!\cdots\!95}{17\!\cdots\!72}a^{12}+\frac{31\!\cdots\!81}{18\!\cdots\!52}a^{11}+\frac{84\!\cdots\!35}{36\!\cdots\!04}a^{10}-\frac{19\!\cdots\!29}{45\!\cdots\!88}a^{9}+\frac{19\!\cdots\!25}{36\!\cdots\!04}a^{8}+\frac{20\!\cdots\!09}{36\!\cdots\!04}a^{7}-\frac{51\!\cdots\!53}{18\!\cdots\!52}a^{6}-\frac{18\!\cdots\!21}{36\!\cdots\!04}a^{5}+\frac{12\!\cdots\!09}{18\!\cdots\!52}a^{4}+\frac{37\!\cdots\!59}{45\!\cdots\!88}a^{3}-\frac{11\!\cdots\!09}{45\!\cdots\!88}a^{2}-\frac{544897021097933}{22\!\cdots\!44}a+\frac{33\!\cdots\!47}{16\!\cdots\!54}$
| 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: | \( 15406380217.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 15406380217.0 \cdot 1}{2\cdot\sqrt{1823491470737247969166873376199879779}}\cr\approx \mathstrut & 6.87428123232 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 - 11*x^22 + 47*x^21 - 78*x^20 - 51*x^19 + 367*x^18 - 359*x^17 - 364*x^16 + 1182*x^15 - 712*x^14 - 1586*x^13 + 3310*x^12 - 393*x^11 - 5271*x^10 + 5059*x^9 + 3246*x^8 - 8091*x^7 - 125*x^6 + 10047*x^5 - 6818*x^4 - 2608*x^3 + 2872*x^2 + 1488*x - 2176)
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^23 - 11*x^22 + 47*x^21 - 78*x^20 - 51*x^19 + 367*x^18 - 359*x^17 - 364*x^16 + 1182*x^15 - 712*x^14 - 1586*x^13 + 3310*x^12 - 393*x^11 - 5271*x^10 + 5059*x^9 + 3246*x^8 - 8091*x^7 - 125*x^6 + 10047*x^5 - 6818*x^4 - 2608*x^3 + 2872*x^2 + 1488*x - 2176, 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^23 - 11*x^22 + 47*x^21 - 78*x^20 - 51*x^19 + 367*x^18 - 359*x^17 - 364*x^16 + 1182*x^15 - 712*x^14 - 1586*x^13 + 3310*x^12 - 393*x^11 - 5271*x^10 + 5059*x^9 + 3246*x^8 - 8091*x^7 - 125*x^6 + 10047*x^5 - 6818*x^4 - 2608*x^3 + 2872*x^2 + 1488*x - 2176);
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^23 - 11*x^22 + 47*x^21 - 78*x^20 - 51*x^19 + 367*x^18 - 359*x^17 - 364*x^16 + 1182*x^15 - 712*x^14 - 1586*x^13 + 3310*x^12 - 393*x^11 - 5271*x^10 + 5059*x^9 + 3246*x^8 - 8091*x^7 - 125*x^6 + 10047*x^5 - 6818*x^4 - 2608*x^3 + 2872*x^2 + 1488*x - 2176);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 46 |
| The 13 conjugacy class representatives for $D_{23}$ |
| Character table for $D_{23}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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
| Galois closure: | 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 | ${\href{/padicField/2.2.0.1}{2} }^{11}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $23$ | $23$ | $23$ | $23$ | $23$ | ${\href{/padicField/17.2.0.1}{2} }^{11}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $23$ | ${\href{/padicField/23.2.0.1}{2} }^{11}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $23$ | $23$ | ${\href{/padicField/37.2.0.1}{2} }^{11}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{11}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{11}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $23$ | ${\href{/padicField/53.2.0.1}{2} }^{11}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $23$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(1979\)
| $\Q_{1979}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |