Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 117*x^19 - 78*x^18 + 5481*x^17 + 7308*x^16 - 128865*x^15 - 262602*x^14 + 1530063*x^13 + 4508112*x^12 - 7197903*x^11 - 37128834*x^10 - 13019517*x^9 + 120573468*x^8 + 200437749*x^7 + 9686586*x^6 - 322807356*x^5 - 455863032*x^4 - 320959248*x^3 - 129747744*x^2 - 28832832*x - 2745984)
gp: K = bnfinit(y^21 - 117*y^19 - 78*y^18 + 5481*y^17 + 7308*y^16 - 128865*y^15 - 262602*y^14 + 1530063*y^13 + 4508112*y^12 - 7197903*y^11 - 37128834*y^10 - 13019517*y^9 + 120573468*y^8 + 200437749*y^7 + 9686586*y^6 - 322807356*y^5 - 455863032*y^4 - 320959248*y^3 - 129747744*y^2 - 28832832*y - 2745984, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 117*x^19 - 78*x^18 + 5481*x^17 + 7308*x^16 - 128865*x^15 - 262602*x^14 + 1530063*x^13 + 4508112*x^12 - 7197903*x^11 - 37128834*x^10 - 13019517*x^9 + 120573468*x^8 + 200437749*x^7 + 9686586*x^6 - 322807356*x^5 - 455863032*x^4 - 320959248*x^3 - 129747744*x^2 - 28832832*x - 2745984);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 117*x^19 - 78*x^18 + 5481*x^17 + 7308*x^16 - 128865*x^15 - 262602*x^14 + 1530063*x^13 + 4508112*x^12 - 7197903*x^11 - 37128834*x^10 - 13019517*x^9 + 120573468*x^8 + 200437749*x^7 + 9686586*x^6 - 322807356*x^5 - 455863032*x^4 - 320959248*x^3 - 129747744*x^2 - 28832832*x - 2745984)
\( x^{21} - 117 x^{19} - 78 x^{18} + 5481 x^{17} + 7308 x^{16} - 128865 x^{15} - 262602 x^{14} + \cdots - 2745984 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[19, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-52483781054437666614504054423280523869246083235156328448\)
\(\medspace = -\,2^{39}\cdot 3^{41}\cdot 13^{15}\cdot 7151^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(450.13\) | 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: |
\(2\), \(3\), \(13\), \(7151\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-78}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{16}a^{8}-\frac{1}{16}a^{7}+\frac{1}{16}a^{6}+\frac{1}{16}a^{5}+\frac{1}{16}a^{4}+\frac{1}{16}a^{3}+\frac{7}{16}a^{2}+\frac{7}{16}a-\frac{1}{8}$, $\frac{1}{16}a^{9}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{2}+\frac{1}{16}a+\frac{3}{8}$, $\frac{1}{16}a^{10}-\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{7}{16}a^{2}-\frac{1}{4}$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{1}{8}a^{4}-\frac{7}{32}a^{3}-\frac{5}{32}a^{2}-\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{10}-\frac{1}{16}a^{7}+\frac{1}{16}a^{5}-\frac{1}{32}a^{4}+\frac{1}{16}a^{3}-\frac{15}{32}a^{2}+\frac{7}{16}a$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{64}a^{11}+\frac{1}{64}a^{10}-\frac{1}{32}a^{9}-\frac{1}{32}a^{8}+\frac{1}{32}a^{7}-\frac{1}{32}a^{6}-\frac{7}{64}a^{5}-\frac{1}{64}a^{4}-\frac{1}{64}a^{3}-\frac{31}{64}a^{2}+\frac{1}{8}a-\frac{7}{16}$, $\frac{1}{64}a^{14}+\frac{1}{64}a^{10}-\frac{1}{16}a^{7}+\frac{7}{64}a^{6}+\frac{1}{16}a^{5}-\frac{3}{16}a^{4}+\frac{1}{16}a^{3}+\frac{23}{64}a^{2}+\frac{7}{16}a+\frac{3}{16}$, $\frac{1}{256}a^{15}-\frac{1}{256}a^{14}-\frac{1}{128}a^{13}-\frac{1}{128}a^{12}-\frac{1}{256}a^{11}+\frac{1}{256}a^{10}-\frac{1}{64}a^{9}+\frac{15}{256}a^{7}-\frac{15}{256}a^{6}+\frac{15}{128}a^{5}-\frac{5}{128}a^{4}+\frac{49}{256}a^{3}-\frac{49}{256}a^{2}+\frac{13}{32}a-\frac{29}{64}$, $\frac{1}{256}a^{16}+\frac{1}{256}a^{14}+\frac{1}{256}a^{12}-\frac{1}{64}a^{11}-\frac{3}{256}a^{10}+\frac{1}{64}a^{9}+\frac{7}{256}a^{8}+\frac{1}{32}a^{7}+\frac{3}{256}a^{6}+\frac{3}{32}a^{5}-\frac{53}{256}a^{4}-\frac{1}{64}a^{3}+\frac{63}{256}a^{2}-\frac{7}{64}a-\frac{5}{64}$, $\frac{1}{4096}a^{17}-\frac{1}{512}a^{16}+\frac{3}{4096}a^{15}+\frac{5}{2048}a^{14}+\frac{1}{4096}a^{13}+\frac{5}{1024}a^{12}-\frac{25}{4096}a^{11}+\frac{7}{2048}a^{10}-\frac{121}{4096}a^{9}+\frac{13}{512}a^{8}-\frac{183}{4096}a^{7}+\frac{115}{2048}a^{6}-\frac{373}{4096}a^{5}+\frac{253}{1024}a^{4}+\frac{781}{4096}a^{3}+\frac{961}{2048}a^{2}-\frac{117}{1024}a-\frac{237}{512}$, $\frac{1}{65536}a^{18}+\frac{3}{32768}a^{17}-\frac{93}{65536}a^{16}+\frac{17}{16384}a^{15}-\frac{371}{65536}a^{14}+\frac{225}{32768}a^{13}-\frac{849}{65536}a^{12}-\frac{11}{2048}a^{11}+\frac{1899}{65536}a^{10}-\frac{731}{32768}a^{9}+\frac{233}{65536}a^{8}-\frac{523}{16384}a^{7}-\frac{7713}{65536}a^{6}+\frac{3415}{32768}a^{5}-\frac{12747}{65536}a^{4}+\frac{937}{8192}a^{3}-\frac{659}{8192}a^{2}-\frac{313}{1024}a-\frac{755}{4096}$, $\frac{1}{1048576}a^{19}+\frac{1}{262144}a^{18}-\frac{105}{1048576}a^{17}-\frac{257}{524288}a^{16}-\frac{251}{1048576}a^{15}-\frac{491}{131072}a^{14}-\frac{4309}{1048576}a^{13}-\frac{5087}{524288}a^{12}+\frac{3371}{1048576}a^{11}-\frac{3235}{262144}a^{10}-\frac{5035}{1048576}a^{9}-\frac{10111}{524288}a^{8}-\frac{34505}{1048576}a^{7}-\frac{7697}{65536}a^{6}-\frac{55591}{1048576}a^{5}-\frac{104721}{524288}a^{4}-\frac{837}{131072}a^{3}-\frac{23473}{65536}a^{2}-\frac{28523}{65536}a-\frac{6605}{32768}$, $\frac{1}{16777216}a^{20}+\frac{1}{8388608}a^{19}-\frac{113}{16777216}a^{18}-\frac{19}{1048576}a^{17}+\frac{4873}{16777216}a^{16}+\frac{8527}{8388608}a^{15}-\frac{94757}{16777216}a^{14}+\frac{18043}{4194304}a^{13}+\frac{101543}{16777216}a^{12}-\frac{3697}{8388608}a^{11}-\frac{396947}{16777216}a^{10}-\frac{21749}{2097152}a^{9}+\frac{263987}{16777216}a^{8}+\frac{257601}{8388608}a^{7}-\frac{907015}{16777216}a^{6}-\frac{522229}{4194304}a^{5}-\frac{350585}{4194304}a^{4}-\frac{31451}{262144}a^{3}-\frac{224777}{1048576}a^{2}+\frac{39375}{262144}a-\frac{11315}{262144}$
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: | $19$ | 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: |
$\frac{84344902695}{2097152}a^{20}-\frac{28562401521}{1048576}a^{19}-\frac{9829665226647}{2097152}a^{18}+\frac{2453063373}{65536}a^{17}+\frac{462241349260767}{2097152}a^{16}+\frac{151663684967505}{1048576}a^{15}-\frac{11\!\cdots\!15}{2097152}a^{14}-\frac{36\!\cdots\!03}{524288}a^{13}+\frac{13\!\cdots\!93}{2097152}a^{12}+\frac{14\!\cdots\!73}{1048576}a^{11}-\frac{80\!\cdots\!09}{2097152}a^{10}-\frac{32\!\cdots\!37}{262144}a^{9}+\frac{65\!\cdots\!45}{2097152}a^{8}+\frac{48\!\cdots\!55}{1048576}a^{7}+\frac{10\!\cdots\!19}{2097152}a^{6}-\frac{15\!\cdots\!67}{524288}a^{5}-\frac{57\!\cdots\!35}{524288}a^{4}-\frac{35\!\cdots\!39}{32768}a^{3}-\frac{72\!\cdots\!51}{131072}a^{2}-\frac{48\!\cdots\!87}{32768}a-\frac{53\!\cdots\!29}{32768}$, $\frac{7264025699031}{16777216}a^{20}-\frac{2461265073801}{8388608}a^{19}-\frac{846555212637415}{16777216}a^{18}+\frac{442593404343}{1048576}a^{17}+\frac{39\!\cdots\!19}{16777216}a^{16}+\frac{13\!\cdots\!69}{8388608}a^{15}-\frac{95\!\cdots\!63}{16777216}a^{14}-\frac{31\!\cdots\!67}{4194304}a^{13}+\frac{11\!\cdots\!77}{16777216}a^{12}+\frac{12\!\cdots\!25}{8388608}a^{11}-\frac{68\!\cdots\!33}{16777216}a^{10}-\frac{27\!\cdots\!03}{2097152}a^{9}+\frac{56\!\cdots\!69}{16777216}a^{8}+\frac{41\!\cdots\!27}{8388608}a^{7}+\frac{88\!\cdots\!59}{16777216}a^{6}-\frac{13\!\cdots\!19}{4194304}a^{5}-\frac{49\!\cdots\!91}{4194304}a^{4}-\frac{30\!\cdots\!49}{262144}a^{3}-\frac{62\!\cdots\!39}{1048576}a^{2}-\frac{41\!\cdots\!79}{262144}a-\frac{45\!\cdots\!41}{262144}$, $\frac{14932960659}{1048576}a^{20}-\frac{5035757769}{524288}a^{19}-\frac{1740363951267}{1048576}a^{18}+\frac{1126950165}{131072}a^{17}+\frac{81841508549883}{1048576}a^{16}+\frac{26966078903889}{524288}a^{15}-\frac{19\!\cdots\!43}{1048576}a^{14}-\frac{649756437228873}{262144}a^{13}+\frac{24\!\cdots\!73}{1048576}a^{12}+\frac{25\!\cdots\!93}{524288}a^{11}-\frac{14\!\cdots\!49}{1048576}a^{10}-\frac{28\!\cdots\!89}{65536}a^{9}+\frac{11\!\cdots\!97}{1048576}a^{8}+\frac{86\!\cdots\!19}{524288}a^{7}+\frac{18\!\cdots\!95}{1048576}a^{6}-\frac{27\!\cdots\!93}{262144}a^{5}-\frac{10\!\cdots\!03}{262144}a^{4}-\frac{31\!\cdots\!31}{8192}a^{3}-\frac{12\!\cdots\!91}{65536}a^{2}-\frac{85\!\cdots\!93}{16384}a-\frac{949943035114133}{16384}$, $\frac{128043170361}{2097152}a^{20}-\frac{43270208343}{1048576}a^{19}-\frac{14922561917961}{2097152}a^{18}+\frac{6145742403}{131072}a^{17}+\frac{701738270045217}{2097152}a^{16}+\frac{230728114313991}{1048576}a^{15}-\frac{16\!\cdots\!45}{2097152}a^{14}-\frac{55\!\cdots\!09}{524288}a^{13}+\frac{21\!\cdots\!55}{2097152}a^{12}+\frac{21\!\cdots\!47}{1048576}a^{11}-\frac{12\!\cdots\!63}{2097152}a^{10}-\frac{49\!\cdots\!89}{262144}a^{9}+\frac{99\!\cdots\!19}{2097152}a^{8}+\frac{73\!\cdots\!93}{1048576}a^{7}+\frac{15\!\cdots\!09}{2097152}a^{6}-\frac{23\!\cdots\!45}{524288}a^{5}-\frac{87\!\cdots\!09}{524288}a^{4}-\frac{54\!\cdots\!35}{32768}a^{3}-\frac{11\!\cdots\!13}{131072}a^{2}-\frac{73\!\cdots\!41}{32768}a-\frac{81\!\cdots\!15}{32768}$, $\frac{737294695407}{8388608}a^{20}-\frac{250024398165}{4194304}a^{19}-\frac{85924335747231}{8388608}a^{18}+\frac{95838575679}{1048576}a^{17}+\frac{40\!\cdots\!11}{8388608}a^{16}+\frac{13\!\cdots\!69}{4194304}a^{15}-\frac{96\!\cdots\!51}{8388608}a^{14}-\frac{31\!\cdots\!17}{2097152}a^{13}+\frac{12\!\cdots\!17}{8388608}a^{12}+\frac{12\!\cdots\!77}{4194304}a^{11}-\frac{70\!\cdots\!33}{8388608}a^{10}-\frac{88\!\cdots\!69}{32768}a^{9}+\frac{57\!\cdots\!81}{8388608}a^{8}+\frac{42\!\cdots\!43}{4194304}a^{7}+\frac{90\!\cdots\!19}{8388608}a^{6}-\frac{13\!\cdots\!57}{2097152}a^{5}-\frac{50\!\cdots\!87}{2097152}a^{4}-\frac{97\!\cdots\!83}{4096}a^{3}-\frac{63\!\cdots\!07}{524288}a^{2}-\frac{42\!\cdots\!77}{131072}a-\frac{46\!\cdots\!73}{131072}$, $\frac{2800657001457}{16777216}a^{20}-\frac{950918907879}{8388608}a^{19}-\frac{326385379038241}{16777216}a^{18}+\frac{12448248777}{65536}a^{17}+\frac{15\!\cdots\!29}{16777216}a^{16}+\frac{50\!\cdots\!59}{8388608}a^{15}-\frac{36\!\cdots\!77}{16777216}a^{14}-\frac{12\!\cdots\!93}{4194304}a^{13}+\frac{46\!\cdots\!39}{16777216}a^{12}+\frac{47\!\cdots\!83}{8388608}a^{11}-\frac{26\!\cdots\!99}{16777216}a^{10}-\frac{10\!\cdots\!51}{2097152}a^{9}+\frac{21\!\cdots\!19}{16777216}a^{8}+\frac{16\!\cdots\!01}{8388608}a^{7}+\frac{34\!\cdots\!37}{16777216}a^{6}-\frac{51\!\cdots\!57}{4194304}a^{5}-\frac{19\!\cdots\!65}{4194304}a^{4}-\frac{11\!\cdots\!33}{262144}a^{3}-\frac{24\!\cdots\!13}{1048576}a^{2}-\frac{15\!\cdots\!85}{262144}a-\frac{17\!\cdots\!71}{262144}$, $\frac{505456528474343}{16777216}a^{20}-\frac{171794070279617}{8388608}a^{19}-\frac{58\!\cdots\!59}{16777216}a^{18}+\frac{4808134016275}{131072}a^{17}+\frac{27\!\cdots\!51}{16777216}a^{16}+\frac{90\!\cdots\!85}{8388608}a^{15}-\frac{66\!\cdots\!63}{16777216}a^{14}-\frac{21\!\cdots\!03}{4194304}a^{13}+\frac{83\!\cdots\!45}{16777216}a^{12}+\frac{85\!\cdots\!17}{8388608}a^{11}-\frac{48\!\cdots\!21}{16777216}a^{10}-\frac{19\!\cdots\!73}{2097152}a^{9}+\frac{39\!\cdots\!01}{16777216}a^{8}+\frac{29\!\cdots\!67}{8388608}a^{7}+\frac{61\!\cdots\!35}{16777216}a^{6}-\frac{92\!\cdots\!71}{4194304}a^{5}-\frac{34\!\cdots\!55}{4194304}a^{4}-\frac{21\!\cdots\!59}{262144}a^{3}-\frac{43\!\cdots\!55}{1048576}a^{2}-\frac{28\!\cdots\!23}{262144}a-\frac{31\!\cdots\!93}{262144}$, $\frac{8198942729}{4194304}a^{20}-\frac{2783008429}{2097152}a^{19}-\frac{955497666665}{4194304}a^{18}+\frac{2285299335}{1048576}a^{17}+\frac{44932195903977}{4194304}a^{16}+\frac{14707376379065}{2097152}a^{15}-\frac{10\!\cdots\!41}{4194304}a^{14}-\frac{177779736890283}{524288}a^{13}+\frac{13\!\cdots\!79}{4194304}a^{12}+\frac{13\!\cdots\!65}{2097152}a^{11}-\frac{77\!\cdots\!71}{4194304}a^{10}-\frac{62\!\cdots\!69}{1048576}a^{9}+\frac{64\!\cdots\!43}{4194304}a^{8}+\frac{47\!\cdots\!99}{2097152}a^{7}+\frac{10\!\cdots\!65}{4194304}a^{6}-\frac{46\!\cdots\!03}{32768}a^{5}-\frac{55\!\cdots\!71}{1048576}a^{4}-\frac{69\!\cdots\!11}{131072}a^{3}-\frac{70\!\cdots\!37}{262144}a^{2}-\frac{292356846528091}{4096}a-\frac{518194319692157}{65536}$, $\frac{2768049316639}{16777216}a^{20}-\frac{936651279265}{8388608}a^{19}-\frac{322593993168303}{16777216}a^{18}+\frac{150660612011}{1048576}a^{17}+\frac{15\!\cdots\!79}{16777216}a^{16}+\frac{49\!\cdots\!57}{8388608}a^{15}-\frac{36\!\cdots\!19}{16777216}a^{14}-\frac{12\!\cdots\!15}{4194304}a^{13}+\frac{45\!\cdots\!93}{16777216}a^{12}+\frac{46\!\cdots\!49}{8388608}a^{11}-\frac{26\!\cdots\!57}{16777216}a^{10}-\frac{10\!\cdots\!95}{2097152}a^{9}+\frac{21\!\cdots\!81}{16777216}a^{8}+\frac{15\!\cdots\!35}{8388608}a^{7}+\frac{33\!\cdots\!75}{16777216}a^{6}-\frac{50\!\cdots\!35}{4194304}a^{5}-\frac{18\!\cdots\!91}{4194304}a^{4}-\frac{11\!\cdots\!65}{262144}a^{3}-\frac{23\!\cdots\!95}{1048576}a^{2}-\frac{15\!\cdots\!87}{262144}a-\frac{17\!\cdots\!65}{262144}$, $\frac{14994481256375}{4194304}a^{20}-\frac{5081019396831}{2097152}a^{19}-\frac{17\!\cdots\!23}{4194304}a^{18}+\frac{3680383830479}{1048576}a^{17}+\frac{82\!\cdots\!91}{4194304}a^{16}+\frac{26\!\cdots\!03}{2097152}a^{15}-\frac{19\!\cdots\!87}{4194304}a^{14}-\frac{16\!\cdots\!75}{262144}a^{13}+\frac{24\!\cdots\!61}{4194304}a^{12}+\frac{25\!\cdots\!39}{2097152}a^{11}-\frac{14\!\cdots\!41}{4194304}a^{10}-\frac{11\!\cdots\!61}{1048576}a^{9}+\frac{11\!\cdots\!57}{4194304}a^{8}+\frac{86\!\cdots\!33}{2097152}a^{7}+\frac{18\!\cdots\!59}{4194304}a^{6}-\frac{13\!\cdots\!15}{524288}a^{5}-\frac{10\!\cdots\!97}{1048576}a^{4}-\frac{12\!\cdots\!71}{131072}a^{3}-\frac{12\!\cdots\!91}{262144}a^{2}-\frac{42\!\cdots\!99}{32768}a-\frac{94\!\cdots\!43}{65536}$, $\frac{20\!\cdots\!51}{16777216}a^{20}-\frac{686427357759265}{8388608}a^{19}-\frac{23\!\cdots\!59}{16777216}a^{18}+\frac{108753264400425}{1048576}a^{17}+\frac{11\!\cdots\!15}{16777216}a^{16}+\frac{36\!\cdots\!33}{8388608}a^{15}-\frac{26\!\cdots\!71}{16777216}a^{14}-\frac{88\!\cdots\!63}{4194304}a^{13}+\frac{33\!\cdots\!21}{16777216}a^{12}+\frac{34\!\cdots\!69}{8388608}a^{11}-\frac{19\!\cdots\!85}{16777216}a^{10}-\frac{77\!\cdots\!83}{2097152}a^{9}+\frac{15\!\cdots\!29}{16777216}a^{8}+\frac{11\!\cdots\!59}{8388608}a^{7}+\frac{24\!\cdots\!47}{16777216}a^{6}-\frac{37\!\cdots\!31}{4194304}a^{5}-\frac{13\!\cdots\!11}{4194304}a^{4}-\frac{85\!\cdots\!81}{262144}a^{3}-\frac{17\!\cdots\!43}{1048576}a^{2}-\frac{11\!\cdots\!27}{262144}a-\frac{12\!\cdots\!77}{262144}$, $\frac{16831420857}{16777216}a^{20}-\frac{5934341959}{8388608}a^{19}-\frac{1960905498025}{16777216}a^{18}+\frac{4367526725}{1048576}a^{17}+\frac{92203563939201}{16777216}a^{16}+\frac{28993383933463}{8388608}a^{15}-\frac{22\!\cdots\!49}{16777216}a^{14}-\frac{715420003257629}{4194304}a^{13}+\frac{27\!\cdots\!19}{16777216}a^{12}+\frac{28\!\cdots\!03}{8388608}a^{11}-\frac{16\!\cdots\!67}{16777216}a^{10}-\frac{63\!\cdots\!09}{2097152}a^{9}+\frac{14\!\cdots\!67}{16777216}a^{8}+\frac{96\!\cdots\!93}{8388608}a^{7}+\frac{20\!\cdots\!73}{16777216}a^{6}-\frac{31\!\cdots\!01}{4194304}a^{5}-\frac{11\!\cdots\!01}{4194304}a^{4}-\frac{69\!\cdots\!63}{262144}a^{3}-\frac{14\!\cdots\!17}{1048576}a^{2}-\frac{93\!\cdots\!93}{262144}a-\frac{10\!\cdots\!63}{262144}$, $\frac{2674576151039}{8388608}a^{20}-\frac{900218104941}{4194304}a^{19}-\frac{311713564373615}{8388608}a^{18}+\frac{152315648409}{1048576}a^{17}+\frac{14\!\cdots\!07}{8388608}a^{16}+\frac{48\!\cdots\!53}{4194304}a^{15}-\frac{35\!\cdots\!47}{8388608}a^{14}-\frac{11\!\cdots\!25}{2097152}a^{13}+\frac{44\!\cdots\!13}{8388608}a^{12}+\frac{45\!\cdots\!65}{4194304}a^{11}-\frac{25\!\cdots\!09}{8388608}a^{10}-\frac{51\!\cdots\!33}{524288}a^{9}+\frac{20\!\cdots\!65}{8388608}a^{8}+\frac{15\!\cdots\!47}{4194304}a^{7}+\frac{32\!\cdots\!51}{8388608}a^{6}-\frac{48\!\cdots\!33}{2097152}a^{5}-\frac{18\!\cdots\!67}{2097152}a^{4}-\frac{56\!\cdots\!67}{65536}a^{3}-\frac{23\!\cdots\!51}{524288}a^{2}-\frac{15\!\cdots\!57}{131072}a-\frac{17\!\cdots\!57}{131072}$, $\frac{751166725626277}{16777216}a^{20}-\frac{254507463221659}{8388608}a^{19}-\frac{87\!\cdots\!05}{16777216}a^{18}+\frac{45625604344649}{1048576}a^{17}+\frac{41\!\cdots\!69}{16777216}a^{16}+\frac{13\!\cdots\!59}{8388608}a^{15}-\frac{98\!\cdots\!73}{16777216}a^{14}-\frac{32\!\cdots\!17}{4194304}a^{13}+\frac{12\!\cdots\!91}{16777216}a^{12}+\frac{12\!\cdots\!63}{8388608}a^{11}-\frac{71\!\cdots\!83}{16777216}a^{10}-\frac{28\!\cdots\!89}{2097152}a^{9}+\frac{58\!\cdots\!59}{16777216}a^{8}+\frac{43\!\cdots\!65}{8388608}a^{7}+\frac{91\!\cdots\!85}{16777216}a^{6}-\frac{13\!\cdots\!17}{4194304}a^{5}-\frac{51\!\cdots\!05}{4194304}a^{4}-\frac{31\!\cdots\!47}{262144}a^{3}-\frac{64\!\cdots\!53}{1048576}a^{2}-\frac{42\!\cdots\!29}{262144}a-\frac{47\!\cdots\!99}{262144}$, $\frac{156751397349}{8388608}a^{20}-\frac{53161449739}{4194304}a^{19}-\frac{18267794271381}{8388608}a^{18}+\frac{10265655919}{524288}a^{17}+\frac{859042830311117}{8388608}a^{16}+\frac{281429600735739}{4194304}a^{15}-\frac{20\!\cdots\!89}{8388608}a^{14}-\frac{68\!\cdots\!09}{2097152}a^{13}+\frac{25\!\cdots\!51}{8388608}a^{12}+\frac{26\!\cdots\!79}{4194304}a^{11}-\frac{14\!\cdots\!95}{8388608}a^{10}-\frac{60\!\cdots\!97}{1048576}a^{9}+\frac{12\!\cdots\!63}{8388608}a^{8}+\frac{90\!\cdots\!57}{4194304}a^{7}+\frac{19\!\cdots\!93}{8388608}a^{6}-\frac{28\!\cdots\!37}{2097152}a^{5}-\frac{10\!\cdots\!97}{2097152}a^{4}-\frac{66\!\cdots\!11}{131072}a^{3}-\frac{13\!\cdots\!05}{524288}a^{2}-\frac{89\!\cdots\!45}{131072}a-\frac{99\!\cdots\!75}{131072}$, $\frac{128065872323421}{16777216}a^{20}-\frac{43391219726499}{8388608}a^{19}-\frac{14\!\cdots\!33}{16777216}a^{18}+\frac{7784835895865}{1048576}a^{17}+\frac{70\!\cdots\!05}{16777216}a^{16}+\frac{23\!\cdots\!91}{8388608}a^{15}-\frac{16\!\cdots\!65}{16777216}a^{14}-\frac{55\!\cdots\!57}{4194304}a^{13}+\frac{21\!\cdots\!91}{16777216}a^{12}+\frac{21\!\cdots\!87}{8388608}a^{11}-\frac{12\!\cdots\!43}{16777216}a^{10}-\frac{49\!\cdots\!61}{2097152}a^{9}+\frac{99\!\cdots\!51}{16777216}a^{8}+\frac{73\!\cdots\!81}{8388608}a^{7}+\frac{15\!\cdots\!49}{16777216}a^{6}-\frac{23\!\cdots\!37}{4194304}a^{5}-\frac{87\!\cdots\!61}{4194304}a^{4}-\frac{54\!\cdots\!63}{262144}a^{3}-\frac{11\!\cdots\!29}{1048576}a^{2}-\frac{73\!\cdots\!21}{262144}a-\frac{81\!\cdots\!75}{262144}$, $\frac{111510554623}{524288}a^{20}-\frac{75347950521}{524288}a^{19}-\frac{12995822860803}{524288}a^{18}+\frac{83486585903}{524288}a^{17}+\frac{611133025905821}{524288}a^{16}+\frac{401975015937263}{524288}a^{15}-\frac{14\!\cdots\!23}{524288}a^{14}-\frac{19\!\cdots\!25}{524288}a^{13}+\frac{18\!\cdots\!39}{524288}a^{12}+\frac{37\!\cdots\!81}{524288}a^{11}-\frac{10\!\cdots\!33}{524288}a^{10}-\frac{34\!\cdots\!91}{524288}a^{9}+\frac{86\!\cdots\!83}{524288}a^{8}+\frac{12\!\cdots\!25}{524288}a^{7}+\frac{13\!\cdots\!51}{524288}a^{6}-\frac{81\!\cdots\!15}{524288}a^{5}-\frac{15\!\cdots\!35}{262144}a^{4}-\frac{37\!\cdots\!65}{65536}a^{3}-\frac{24\!\cdots\!15}{8192}a^{2}-\frac{25\!\cdots\!91}{32768}a-\frac{14\!\cdots\!99}{16384}$, $\frac{16502031500229}{8388608}a^{20}-\frac{5591229094263}{4194304}a^{19}-\frac{19\!\cdots\!05}{8388608}a^{18}+\frac{2006773909801}{1048576}a^{17}+\frac{90\!\cdots\!49}{8388608}a^{16}+\frac{29\!\cdots\!15}{4194304}a^{15}-\frac{21\!\cdots\!61}{8388608}a^{14}-\frac{71\!\cdots\!99}{2097152}a^{13}+\frac{27\!\cdots\!39}{8388608}a^{12}+\frac{27\!\cdots\!55}{4194304}a^{11}-\frac{15\!\cdots\!71}{8388608}a^{10}-\frac{15\!\cdots\!93}{262144}a^{9}+\frac{12\!\cdots\!71}{8388608}a^{8}+\frac{95\!\cdots\!53}{4194304}a^{7}+\frac{20\!\cdots\!13}{8388608}a^{6}-\frac{30\!\cdots\!07}{2097152}a^{5}-\frac{11\!\cdots\!01}{2097152}a^{4}-\frac{17\!\cdots\!55}{32768}a^{3}-\frac{14\!\cdots\!37}{524288}a^{2}-\frac{94\!\cdots\!27}{131072}a-\frac{10\!\cdots\!43}{131072}$, $\frac{2658099227915}{2097152}a^{20}-\frac{914141686617}{1048576}a^{19}-\frac{309740106783899}{2097152}a^{18}+\frac{714063886631}{262144}a^{17}+\frac{14\!\cdots\!83}{2097152}a^{16}+\frac{47\!\cdots\!69}{1048576}a^{15}-\frac{34\!\cdots\!27}{2097152}a^{14}-\frac{11\!\cdots\!53}{524288}a^{13}+\frac{43\!\cdots\!57}{2097152}a^{12}+\frac{44\!\cdots\!97}{1048576}a^{11}-\frac{25\!\cdots\!37}{2097152}a^{10}-\frac{25\!\cdots\!71}{65536}a^{9}+\frac{21\!\cdots\!77}{2097152}a^{8}+\frac{15\!\cdots\!07}{1048576}a^{7}+\frac{32\!\cdots\!71}{2097152}a^{6}-\frac{49\!\cdots\!13}{524288}a^{5}-\frac{18\!\cdots\!51}{524288}a^{4}-\frac{13\!\cdots\!03}{4096}a^{3}-\frac{22\!\cdots\!75}{131072}a^{2}-\frac{14\!\cdots\!57}{32768}a-\frac{16\!\cdots\!37}{32768}$
| 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: | \( 3949442615580000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{19}\cdot(2\pi)^{1}\cdot 3949442615580000000000 \cdot 1}{2\cdot\sqrt{52483781054437666614504054423280523869246083235156328448}}\cr\approx \mathstrut & 0.897931156027301 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 117*x^19 - 78*x^18 + 5481*x^17 + 7308*x^16 - 128865*x^15 - 262602*x^14 + 1530063*x^13 + 4508112*x^12 - 7197903*x^11 - 37128834*x^10 - 13019517*x^9 + 120573468*x^8 + 200437749*x^7 + 9686586*x^6 - 322807356*x^5 - 455863032*x^4 - 320959248*x^3 - 129747744*x^2 - 28832832*x - 2745984)
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^21 - 117*x^19 - 78*x^18 + 5481*x^17 + 7308*x^16 - 128865*x^15 - 262602*x^14 + 1530063*x^13 + 4508112*x^12 - 7197903*x^11 - 37128834*x^10 - 13019517*x^9 + 120573468*x^8 + 200437749*x^7 + 9686586*x^6 - 322807356*x^5 - 455863032*x^4 - 320959248*x^3 - 129747744*x^2 - 28832832*x - 2745984, 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^21 - 117*x^19 - 78*x^18 + 5481*x^17 + 7308*x^16 - 128865*x^15 - 262602*x^14 + 1530063*x^13 + 4508112*x^12 - 7197903*x^11 - 37128834*x^10 - 13019517*x^9 + 120573468*x^8 + 200437749*x^7 + 9686586*x^6 - 322807356*x^5 - 455863032*x^4 - 320959248*x^3 - 129747744*x^2 - 28832832*x - 2745984);
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^21 - 117*x^19 - 78*x^18 + 5481*x^17 + 7308*x^16 - 128865*x^15 - 262602*x^14 + 1530063*x^13 + 4508112*x^12 - 7197903*x^11 - 37128834*x^10 - 13019517*x^9 + 120573468*x^8 + 200437749*x^7 + 9686586*x^6 - 322807356*x^5 - 455863032*x^4 - 320959248*x^3 - 129747744*x^2 - 28832832*x - 2745984);
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
$C_3^7.C_2\wr F_7$ (as 21T142):
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 11757312 |
| The 168 conjugacy class representatives for $C_3^7.C_2\wr F_7$ |
| Character table for $C_3^7.C_2\wr F_7$ |
Intermediate fields
| 7.7.138584369664.1 |
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 42 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 | R | R | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | $18{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{5}$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{5}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
| 2.12.30.63 | $x^{12} + 4 x^{11} + 40 x^{10} + 504 x^{9} + 1750 x^{8} + 1648 x^{7} + 3328 x^{6} + 11296 x^{5} + 3964 x^{4} + 12624 x^{3} + 7200 x^{2} + 4320 x + 4072$ | $4$ | $3$ | $30$ | 12T87 | $[2, 2, 2, 3, 7/2, 7/2]^{3}$ | |
|
\(3\)
| Deg $21$ | $21$ | $1$ | $41$ | |||
|
\(13\)
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 13.12.10.1 | $x^{12} + 72 x^{11} + 2172 x^{10} + 35280 x^{9} + 328380 x^{8} + 1703232 x^{7} + 4282170 x^{6} + 3407400 x^{5} + 1340820 x^{4} + 712800 x^{3} + 3855192 x^{2} + 18082080 x + 35650393$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
|
\(7151\)
| $\Q_{7151}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7151}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7151}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7151}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ |