Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 74*x^20 + 2449*x^18 - 47772*x^16 + 608919*x^14 - 5307720*x^12 + 32127975*x^10 - 134044248*x^8 + 373194336*x^6 - 645946216*x^4 + 597000440*x^2 - 195030400)
gp: K = bnfinit(y^22 - 74*y^20 + 2449*y^18 - 47772*y^16 + 608919*y^14 - 5307720*y^12 + 32127975*y^10 - 134044248*y^8 + 373194336*y^6 - 645946216*y^4 + 597000440*y^2 - 195030400, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 74*x^20 + 2449*x^18 - 47772*x^16 + 608919*x^14 - 5307720*x^12 + 32127975*x^10 - 134044248*x^8 + 373194336*x^6 - 645946216*x^4 + 597000440*x^2 - 195030400);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 74*x^20 + 2449*x^18 - 47772*x^16 + 608919*x^14 - 5307720*x^12 + 32127975*x^10 - 134044248*x^8 + 373194336*x^6 - 645946216*x^4 + 597000440*x^2 - 195030400)
\( x^{22} - 74 x^{20} + 2449 x^{18} - 47772 x^{16} + 608919 x^{14} - 5307720 x^{12} + 32127975 x^{10} + \cdots - 195030400 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[14, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(12690240639024486099207064421849753745845788293805139479005039009036081561600\)
\(\medspace = 2^{45}\cdot 3^{28}\cdot 5^{2}\cdot 7^{4}\cdot 23^{4}\cdot 59\cdot 137^{16}\cdot 1033\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(2879.05\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\), \(23\), \(59\), \(137\), \(1033\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{121894}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{12}a^{18}-\frac{1}{4}a^{14}-\frac{1}{2}a^{12}+\frac{1}{4}a^{10}-\frac{1}{2}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{3}$, $\frac{1}{12}a^{19}-\frac{1}{4}a^{15}-\frac{1}{2}a^{13}+\frac{1}{4}a^{11}-\frac{1}{2}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{3}a$, $\frac{1}{27000}a^{20}-\frac{527}{13500}a^{18}-\frac{153}{1000}a^{16}+\frac{64}{375}a^{14}-\frac{1807}{9000}a^{12}+\frac{9}{50}a^{10}+\frac{1}{40}a^{8}-\frac{229}{2250}a^{6}+\frac{191}{750}a^{4}+\frac{1663}{3375}a^{2}+\frac{163}{675}$, $\frac{1}{216000}a^{21}-\frac{1}{54000}a^{20}+\frac{1723}{108000}a^{19}+\frac{527}{27000}a^{18}+\frac{1347}{8000}a^{17}-\frac{347}{2000}a^{16}-\frac{247}{6000}a^{15}-\frac{32}{375}a^{14}-\frac{6307}{72000}a^{13}+\frac{6307}{18000}a^{12}-\frac{83}{200}a^{11}-\frac{9}{100}a^{10}-\frac{19}{320}a^{9}+\frac{19}{80}a^{8}-\frac{677}{9000}a^{7}-\frac{2021}{4500}a^{6}-\frac{23}{750}a^{5}+\frac{46}{375}a^{4}+\frac{1663}{27000}a^{3}-\frac{1663}{6750}a^{2}-\frac{1637}{5400}a-\frac{163}{1350}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: | $17$ | 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: |
$\frac{6142481}{13500}a^{20}-\frac{422396099}{13500}a^{18}+\frac{474050157}{500}a^{16}-\frac{24945165353}{1500}a^{14}+\frac{838506985333}{4500}a^{12}-\frac{138635189309}{100}a^{10}+\frac{137622958781}{20}a^{8}-\frac{99923960602721}{4500}a^{6}+\frac{16323256999621}{375}a^{4}-\frac{149976185670194}{3375}a^{2}+\frac{10349732685031}{675}$, $\frac{508699009}{2250}a^{20}-\frac{66275860097}{4500}a^{18}+\frac{52904187247}{125}a^{16}-\frac{3525828786909}{500}a^{14}+\frac{56402287777487}{750}a^{12}-\frac{53380877991231}{100}a^{10}+\frac{25346934348587}{10}a^{8}-\frac{11\!\cdots\!13}{1500}a^{6}+\frac{18\!\cdots\!63}{125}a^{4}-\frac{16\!\cdots\!82}{1125}a^{2}+\frac{11\!\cdots\!93}{225}$, $\frac{363317363}{3375}a^{20}-\frac{96088199783}{13500}a^{18}+\frac{51883665797}{250}a^{16}-\frac{5260255561451}{1500}a^{14}+\frac{42646050392434}{1125}a^{12}-\frac{27255834226903}{100}a^{10}+\frac{6549844725868}{5}a^{8}-\frac{18\!\cdots\!07}{4500}a^{6}+\frac{29\!\cdots\!07}{375}a^{4}-\frac{26\!\cdots\!98}{3375}a^{2}+\frac{17\!\cdots\!27}{675}$, $\frac{15007510961}{13500}a^{20}-\frac{1022971860269}{13500}a^{18}+\frac{1140104496467}{500}a^{16}-\frac{59695764310343}{1500}a^{14}+\frac{20\!\cdots\!23}{4500}a^{12}-\frac{330497039308279}{100}a^{10}+\frac{328430659986771}{20}a^{8}-\frac{23\!\cdots\!51}{4500}a^{6}+\frac{78\!\cdots\!27}{750}a^{4}-\frac{36\!\cdots\!89}{3375}a^{2}+\frac{25\!\cdots\!61}{675}$, $\frac{100471139344}{3375}a^{20}-\frac{18356284339429}{13500}a^{18}+\frac{5353524740111}{250}a^{16}-\frac{73181602490113}{1500}a^{14}-\frac{32\!\cdots\!58}{1125}a^{12}+\frac{44\!\cdots\!11}{100}a^{10}-\frac{16\!\cdots\!66}{5}a^{8}+\frac{62\!\cdots\!59}{4500}a^{6}-\frac{12\!\cdots\!59}{375}a^{4}+\frac{12\!\cdots\!51}{3375}a^{2}-\frac{93\!\cdots\!49}{675}$, $\frac{7774529}{2700}a^{20}-\frac{543876341}{2700}a^{18}+\frac{624001063}{100}a^{16}-\frac{33751889327}{300}a^{14}+\frac{1173249062047}{900}a^{12}-\frac{201941023071}{20}a^{10}+\frac{210236893739}{4}a^{8}-\frac{161388879023339}{900}a^{6}+\frac{56233010651753}{150}a^{4}-\frac{277715094947246}{675}a^{2}+\frac{20562043247299}{135}$, $\frac{25\!\cdots\!41}{13500}a^{20}-\frac{17\!\cdots\!89}{13500}a^{18}+\frac{19\!\cdots\!27}{500}a^{16}-\frac{10\!\cdots\!83}{1500}a^{14}+\frac{33\!\cdots\!63}{4500}a^{12}-\frac{54\!\cdots\!99}{100}a^{10}+\frac{54\!\cdots\!51}{20}a^{8}-\frac{39\!\cdots\!31}{4500}a^{6}+\frac{12\!\cdots\!37}{750}a^{4}-\frac{58\!\cdots\!59}{3375}a^{2}+\frac{40\!\cdots\!91}{675}$, $\frac{70\!\cdots\!61}{36000}a^{21}-\frac{13\!\cdots\!21}{27000}a^{20}-\frac{23\!\cdots\!97}{18000}a^{19}+\frac{44\!\cdots\!67}{13500}a^{18}+\frac{15\!\cdots\!01}{4000}a^{17}-\frac{98\!\cdots\!87}{1000}a^{16}-\frac{68\!\cdots\!67}{1000}a^{15}+\frac{64\!\cdots\!81}{375}a^{14}+\frac{90\!\cdots\!73}{12000}a^{13}-\frac{17\!\cdots\!53}{9000}a^{12}-\frac{55\!\cdots\!39}{100}a^{11}+\frac{69\!\cdots\!61}{50}a^{10}+\frac{43\!\cdots\!23}{160}a^{9}-\frac{27\!\cdots\!41}{40}a^{8}-\frac{13\!\cdots\!47}{1500}a^{7}+\frac{49\!\cdots\!09}{2250}a^{6}+\frac{21\!\cdots\!47}{125}a^{5}-\frac{16\!\cdots\!93}{375}a^{4}-\frac{78\!\cdots\!57}{4500}a^{3}+\frac{14\!\cdots\!27}{3375}a^{2}+\frac{54\!\cdots\!43}{900}a-\frac{10\!\cdots\!98}{675}$, $\frac{70\!\cdots\!61}{36000}a^{21}+\frac{13\!\cdots\!21}{27000}a^{20}-\frac{23\!\cdots\!97}{18000}a^{19}-\frac{44\!\cdots\!67}{13500}a^{18}+\frac{15\!\cdots\!01}{4000}a^{17}+\frac{98\!\cdots\!87}{1000}a^{16}-\frac{68\!\cdots\!67}{1000}a^{15}-\frac{64\!\cdots\!81}{375}a^{14}+\frac{90\!\cdots\!73}{12000}a^{13}+\frac{17\!\cdots\!53}{9000}a^{12}-\frac{55\!\cdots\!39}{100}a^{11}-\frac{69\!\cdots\!61}{50}a^{10}+\frac{43\!\cdots\!23}{160}a^{9}+\frac{27\!\cdots\!41}{40}a^{8}-\frac{13\!\cdots\!47}{1500}a^{7}-\frac{49\!\cdots\!09}{2250}a^{6}+\frac{21\!\cdots\!47}{125}a^{5}+\frac{16\!\cdots\!93}{375}a^{4}-\frac{78\!\cdots\!57}{4500}a^{3}-\frac{14\!\cdots\!27}{3375}a^{2}+\frac{54\!\cdots\!43}{900}a+\frac{10\!\cdots\!98}{675}$, $\frac{11\!\cdots\!91}{43200}a^{21}-\frac{35\!\cdots\!01}{54000}a^{20}-\frac{40\!\cdots\!07}{21600}a^{19}+\frac{12\!\cdots\!77}{27000}a^{18}+\frac{89\!\cdots\!77}{1600}a^{17}-\frac{27\!\cdots\!47}{2000}a^{16}-\frac{11\!\cdots\!77}{1200}a^{15}+\frac{35\!\cdots\!47}{1500}a^{14}+\frac{15\!\cdots\!63}{14400}a^{13}-\frac{47\!\cdots\!93}{18000}a^{12}-\frac{32\!\cdots\!33}{40}a^{11}+\frac{98\!\cdots\!83}{50}a^{10}+\frac{25\!\cdots\!75}{64}a^{9}-\frac{78\!\cdots\!81}{80}a^{8}-\frac{23\!\cdots\!07}{1800}a^{7}+\frac{71\!\cdots\!77}{2250}a^{6}+\frac{38\!\cdots\!57}{150}a^{5}-\frac{46\!\cdots\!83}{750}a^{4}-\frac{14\!\cdots\!67}{5400}a^{3}+\frac{43\!\cdots\!37}{6750}a^{2}+\frac{98\!\cdots\!33}{1080}a-\frac{29\!\cdots\!13}{1350}$, $\frac{90\!\cdots\!99}{36000}a^{21}-\frac{70\!\cdots\!81}{1000}a^{20}-\frac{29\!\cdots\!23}{18000}a^{19}+\frac{11\!\cdots\!81}{250}a^{18}+\frac{19\!\cdots\!59}{4000}a^{17}-\frac{13\!\cdots\!89}{1000}a^{16}-\frac{81\!\cdots\!53}{1000}a^{15}+\frac{11\!\cdots\!51}{500}a^{14}+\frac{10\!\cdots\!07}{12000}a^{13}-\frac{24\!\cdots\!99}{1000}a^{12}-\frac{63\!\cdots\!01}{100}a^{11}+\frac{17\!\cdots\!59}{100}a^{10}+\frac{48\!\cdots\!37}{160}a^{9}-\frac{34\!\cdots\!07}{40}a^{8}-\frac{14\!\cdots\!73}{1500}a^{7}+\frac{13\!\cdots\!19}{500}a^{6}+\frac{22\!\cdots\!48}{125}a^{5}-\frac{63\!\cdots\!32}{125}a^{4}-\frac{82\!\cdots\!13}{4500}a^{3}+\frac{63\!\cdots\!47}{125}a^{2}+\frac{55\!\cdots\!37}{900}a-\frac{43\!\cdots\!28}{25}$, $\frac{55\!\cdots\!13}{216000}a^{21}+\frac{26\!\cdots\!09}{54000}a^{20}-\frac{19\!\cdots\!01}{108000}a^{19}-\frac{93\!\cdots\!93}{27000}a^{18}+\frac{45\!\cdots\!11}{8000}a^{17}+\frac{21\!\cdots\!23}{2000}a^{16}-\frac{61\!\cdots\!11}{6000}a^{15}-\frac{29\!\cdots\!73}{1500}a^{14}+\frac{86\!\cdots\!09}{72000}a^{13}+\frac{40\!\cdots\!37}{18000}a^{12}-\frac{18\!\cdots\!79}{200}a^{11}-\frac{44\!\cdots\!61}{25}a^{10}+\frac{15\!\cdots\!53}{320}a^{9}+\frac{74\!\cdots\!49}{80}a^{8}-\frac{15\!\cdots\!01}{9000}a^{7}-\frac{35\!\cdots\!84}{1125}a^{6}+\frac{13\!\cdots\!88}{375}a^{5}+\frac{50\!\cdots\!47}{750}a^{4}-\frac{10\!\cdots\!81}{27000}a^{3}-\frac{49\!\cdots\!33}{6750}a^{2}+\frac{75\!\cdots\!19}{5400}a+\frac{35\!\cdots\!17}{1350}$, $\frac{36\!\cdots\!23}{8640}a^{21}+\frac{78\!\cdots\!79}{54000}a^{20}-\frac{11\!\cdots\!11}{4320}a^{19}-\frac{24\!\cdots\!83}{27000}a^{18}+\frac{23\!\cdots\!41}{320}a^{17}+\frac{49\!\cdots\!13}{2000}a^{16}-\frac{28\!\cdots\!01}{240}a^{15}-\frac{60\!\cdots\!63}{1500}a^{14}+\frac{35\!\cdots\!59}{2880}a^{13}+\frac{74\!\cdots\!47}{18000}a^{12}-\frac{67\!\cdots\!65}{8}a^{11}-\frac{71\!\cdots\!91}{25}a^{10}+\frac{25\!\cdots\!91}{64}a^{9}+\frac{10\!\cdots\!39}{80}a^{8}-\frac{43\!\cdots\!51}{360}a^{7}-\frac{45\!\cdots\!79}{1125}a^{6}+\frac{33\!\cdots\!63}{15}a^{5}+\frac{56\!\cdots\!07}{750}a^{4}-\frac{23\!\cdots\!11}{1080}a^{3}-\frac{49\!\cdots\!73}{6750}a^{2}+\frac{15\!\cdots\!49}{216}a+\frac{33\!\cdots\!27}{1350}$, $\frac{34\!\cdots\!07}{216000}a^{21}+\frac{86\!\cdots\!21}{18000}a^{20}-\frac{11\!\cdots\!39}{108000}a^{19}-\frac{28\!\cdots\!17}{9000}a^{18}+\frac{23\!\cdots\!29}{8000}a^{17}+\frac{18\!\cdots\!61}{2000}a^{16}-\frac{29\!\cdots\!29}{6000}a^{15}-\frac{74\!\cdots\!37}{500}a^{14}+\frac{37\!\cdots\!51}{72000}a^{13}+\frac{95\!\cdots\!53}{6000}a^{12}-\frac{74\!\cdots\!31}{200}a^{11}-\frac{28\!\cdots\!27}{25}a^{10}+\frac{56\!\cdots\!07}{320}a^{9}+\frac{42\!\cdots\!63}{80}a^{8}-\frac{48\!\cdots\!89}{9000}a^{7}-\frac{61\!\cdots\!46}{375}a^{6}+\frac{37\!\cdots\!07}{375}a^{5}+\frac{38\!\cdots\!34}{125}a^{4}-\frac{26\!\cdots\!59}{27000}a^{3}-\frac{68\!\cdots\!77}{2250}a^{2}+\frac{18\!\cdots\!41}{5400}a+\frac{46\!\cdots\!73}{450}$, $\frac{20\!\cdots\!77}{13500}a^{21}+\frac{19\!\cdots\!99}{500}a^{20}-\frac{13\!\cdots\!83}{13500}a^{19}-\frac{13\!\cdots\!21}{500}a^{18}+\frac{14\!\cdots\!69}{500}a^{17}+\frac{39\!\cdots\!31}{500}a^{16}-\frac{76\!\cdots\!51}{1500}a^{15}-\frac{67\!\cdots\!83}{500}a^{14}+\frac{25\!\cdots\!61}{4500}a^{13}+\frac{74\!\cdots\!71}{500}a^{12}-\frac{40\!\cdots\!03}{100}a^{11}-\frac{10\!\cdots\!97}{100}a^{10}+\frac{39\!\cdots\!17}{20}a^{9}+\frac{10\!\cdots\!63}{20}a^{8}-\frac{28\!\cdots\!07}{4500}a^{7}-\frac{83\!\cdots\!77}{500}a^{6}+\frac{45\!\cdots\!82}{375}a^{5}+\frac{80\!\cdots\!37}{250}a^{4}-\frac{82\!\cdots\!71}{6750}a^{3}-\frac{40\!\cdots\!01}{125}a^{2}+\frac{28\!\cdots\!27}{675}a+\frac{27\!\cdots\!49}{25}$, $\frac{20\!\cdots\!97}{9000}a^{21}+\frac{17\!\cdots\!16}{3375}a^{20}-\frac{70\!\cdots\!19}{4500}a^{19}-\frac{48\!\cdots\!81}{13500}a^{18}+\frac{46\!\cdots\!77}{1000}a^{17}+\frac{26\!\cdots\!29}{250}a^{16}-\frac{10\!\cdots\!92}{125}a^{15}-\frac{27\!\cdots\!57}{1500}a^{14}+\frac{27\!\cdots\!21}{3000}a^{13}+\frac{23\!\cdots\!88}{1125}a^{12}-\frac{34\!\cdots\!31}{50}a^{11}-\frac{15\!\cdots\!21}{100}a^{10}+\frac{13\!\cdots\!11}{40}a^{9}+\frac{37\!\cdots\!46}{5}a^{8}-\frac{83\!\cdots\!13}{750}a^{7}-\frac{10\!\cdots\!49}{4500}a^{6}+\frac{56\!\cdots\!27}{250}a^{5}+\frac{16\!\cdots\!74}{375}a^{4}-\frac{28\!\cdots\!64}{1125}a^{3}-\frac{14\!\cdots\!86}{3375}a^{2}+\frac{23\!\cdots\!36}{225}a+\frac{92\!\cdots\!89}{675}$, $\frac{18\!\cdots\!37}{216000}a^{21}-\frac{12\!\cdots\!73}{54000}a^{20}-\frac{67\!\cdots\!49}{108000}a^{19}+\frac{43\!\cdots\!21}{27000}a^{18}+\frac{15\!\cdots\!39}{8000}a^{17}-\frac{10\!\cdots\!31}{2000}a^{16}-\frac{20\!\cdots\!39}{6000}a^{15}+\frac{13\!\cdots\!81}{1500}a^{14}+\frac{28\!\cdots\!41}{72000}a^{13}-\frac{19\!\cdots\!89}{18000}a^{12}-\frac{61\!\cdots\!71}{200}a^{11}+\frac{20\!\cdots\!67}{25}a^{10}+\frac{49\!\cdots\!17}{320}a^{9}-\frac{33\!\cdots\!73}{80}a^{8}-\frac{46\!\cdots\!49}{9000}a^{7}+\frac{15\!\cdots\!73}{1125}a^{6}+\frac{38\!\cdots\!87}{375}a^{5}-\frac{10\!\cdots\!17}{375}a^{4}-\frac{28\!\cdots\!69}{27000}a^{3}+\frac{19\!\cdots\!01}{6750}a^{2}+\frac{19\!\cdots\!31}{5400}a-\frac{13\!\cdots\!49}{1350}$
| 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: | \( 171825983862000000000000000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 171825983862000000000000000000000 \cdot 1}{2\cdot\sqrt{12690240639024486099207064421849753745845788293805139479005039009036081561600}}\cr\approx \mathstrut & 19.4743659906574 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 74*x^20 + 2449*x^18 - 47772*x^16 + 608919*x^14 - 5307720*x^12 + 32127975*x^10 - 134044248*x^8 + 373194336*x^6 - 645946216*x^4 + 597000440*x^2 - 195030400)
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^22 - 74*x^20 + 2449*x^18 - 47772*x^16 + 608919*x^14 - 5307720*x^12 + 32127975*x^10 - 134044248*x^8 + 373194336*x^6 - 645946216*x^4 + 597000440*x^2 - 195030400, 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^22 - 74*x^20 + 2449*x^18 - 47772*x^16 + 608919*x^14 - 5307720*x^12 + 32127975*x^10 - 134044248*x^8 + 373194336*x^6 - 645946216*x^4 + 597000440*x^2 - 195030400);
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^22 - 74*x^20 + 2449*x^18 - 47772*x^16 + 608919*x^14 - 5307720*x^12 + 32127975*x^10 - 134044248*x^8 + 373194336*x^6 - 645946216*x^4 + 597000440*x^2 - 195030400);
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
$C_2^{11}.A_{11}$ (as 22T52):
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 40874803200 |
| The 400 conjugacy class representatives for $C_2^{11}.A_{11}$ |
| Character table for $C_2^{11}.A_{11}$ |
Intermediate fields
| 11.7.63019333158425674204677255696384.1 |
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 44 sibling: | 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 | R | R | R | R | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | $18{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}$ | R |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.8.21.1 | $x^{8} + 14 x^{6} + 8 x^{4} + 2$ | $8$ | $1$ | $21$ | $C_2 \wr C_2\wr C_2$ | $[2, 2, 3, 7/2, 7/2, 15/4]^{2}$ | |
| 2.12.24.424 | $x^{12} + 2 x^{10} + 4 x^{9} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 6 x^{2} + 4 x + 10$ | $12$ | $1$ | $24$ | 12T149 | $[4/3, 4/3, 2, 7/3, 7/3, 3]_{3}^{2}$ | |
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.9.13.2 | $x^{9} + 6 x^{5} + 3 x^{3} + 3$ | $9$ | $1$ | $13$ | $C_3^2 : D_{6} $ | $[3/2, 3/2, 5/3]_{2}^{2}$ | |
| 3.9.13.2 | $x^{9} + 6 x^{5} + 3 x^{3} + 3$ | $9$ | $1$ | $13$ | $C_3^2 : D_{6} $ | $[3/2, 3/2, 5/3]_{2}^{2}$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.8.0.1 | $x^{8} + x^{4} + 3 x^{2} + 4 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.14.0.1 | $x^{14} + 5 x^{7} + 6 x^{5} + 2 x^{4} + 3 x^{2} + 6 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.14.0.1 | $x^{14} + x^{8} + 5 x^{7} + 16 x^{6} + x^{5} + 18 x^{4} + 19 x^{3} + x^{2} + 22 x + 5$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(59\)
| 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 59.6.0.1 | $x^{6} + 2 x^{4} + 18 x^{3} + 38 x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 59.6.0.1 | $x^{6} + 2 x^{4} + 18 x^{3} + 38 x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 59.8.0.1 | $x^{8} + 16 x^{4} + 32 x^{3} + 2 x^{2} + 50 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(137\)
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 137.5.4.1 | $x^{5} + 137$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 137.10.8.1 | $x^{10} + 655 x^{9} + 171625 x^{8} + 22488770 x^{7} + 1474044185 x^{6} + 38714410755 x^{5} + 4422222290 x^{4} + 225901280 x^{3} + 3082903315 x^{2} + 201591447850 x + 5280771081809$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
|
\(1033\)
| $\Q_{1033}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1033}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |