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Properties

Label 40.0.13813855852...2736.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 11^{36}\cdot 17^{20}$
Root discriminant $71.37$
Ramified primes $2, 11, 17$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1099511627776, 0, -618475290624, 0, 279172874240, 0, -118380036096, 0, 49140465664, 0, -20242759680, 0, 8315273216, 0, -3412168704, 0, 1399640320, 0, -574037136, 0, 235418369, 0, -35877321, 0, 5467345, 0, -833049, 0, 126881, 0, -19305, 0, 2929, 0, -441, 0, 65, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 9*x^38 + 65*x^36 - 441*x^34 + 2929*x^32 - 19305*x^30 + 126881*x^28 - 833049*x^26 + 5467345*x^24 - 35877321*x^22 + 235418369*x^20 - 574037136*x^18 + 1399640320*x^16 - 3412168704*x^14 + 8315273216*x^12 - 20242759680*x^10 + 49140465664*x^8 - 118380036096*x^6 + 279172874240*x^4 - 618475290624*x^2 + 1099511627776)
 
gp: K = bnfinit(x^40 - 9*x^38 + 65*x^36 - 441*x^34 + 2929*x^32 - 19305*x^30 + 126881*x^28 - 833049*x^26 + 5467345*x^24 - 35877321*x^22 + 235418369*x^20 - 574037136*x^18 + 1399640320*x^16 - 3412168704*x^14 + 8315273216*x^12 - 20242759680*x^10 + 49140465664*x^8 - 118380036096*x^6 + 279172874240*x^4 - 618475290624*x^2 + 1099511627776, 1)
 

Normalized defining polynomial

\( x^{40} - 9 x^{38} + 65 x^{36} - 441 x^{34} + 2929 x^{32} - 19305 x^{30} + 126881 x^{28} - 833049 x^{26} + 5467345 x^{24} - 35877321 x^{22} + 235418369 x^{20} - 574037136 x^{18} + 1399640320 x^{16} - 3412168704 x^{14} + 8315273216 x^{12} - 20242759680 x^{10} + 49140465664 x^{8} - 118380036096 x^{6} + 279172874240 x^{4} - 618475290624 x^{2} + 1099511627776 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(138138558520952628335754257563768298595689088682615055246398014526886772736=2^{40}\cdot 11^{36}\cdot 17^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.37$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(748=2^{2}\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{748}(1,·)$, $\chi_{748}(375,·)$, $\chi_{748}(645,·)$, $\chi_{748}(135,·)$, $\chi_{748}(137,·)$, $\chi_{748}(271,·)$, $\chi_{748}(273,·)$, $\chi_{748}(67,·)$, $\chi_{748}(409,·)$, $\chi_{748}(543,·)$, $\chi_{748}(545,·)$, $\chi_{748}(35,·)$, $\chi_{748}(679,·)$, $\chi_{748}(169,·)$, $\chi_{748}(171,·)$, $\chi_{748}(305,·)$, $\chi_{748}(307,·)$, $\chi_{748}(441,·)$, $\chi_{748}(443,·)$, $\chi_{748}(577,·)$, $\chi_{748}(579,·)$, $\chi_{748}(69,·)$, $\chi_{748}(713,·)$, $\chi_{748}(203,·)$, $\chi_{748}(205,·)$, $\chi_{748}(339,·)$, $\chi_{748}(475,·)$, $\chi_{748}(477,·)$, $\chi_{748}(101,·)$, $\chi_{748}(611,·)$, $\chi_{748}(613,·)$, $\chi_{748}(103,·)$, $\chi_{748}(747,·)$, $\chi_{748}(647,·)$, $\chi_{748}(237,·)$, $\chi_{748}(239,·)$, $\chi_{748}(373,·)$, $\chi_{748}(681,·)$, $\chi_{748}(509,·)$, $\chi_{748}(511,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{4} a^{21} - \frac{1}{4} a^{19} + \frac{1}{4} a^{17} - \frac{1}{4} a^{15} + \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{3766693904} a^{22} + \frac{7}{16} a^{20} + \frac{1}{16} a^{18} + \frac{7}{16} a^{16} + \frac{1}{16} a^{14} + \frac{7}{16} a^{12} + \frac{1}{16} a^{10} + \frac{7}{16} a^{8} + \frac{1}{16} a^{6} + \frac{7}{16} a^{4} + \frac{1}{16} a^{2} - \frac{35877321}{235418369}$, $\frac{1}{15066775616} a^{23} + \frac{7}{64} a^{21} - \frac{15}{64} a^{19} + \frac{23}{64} a^{17} - \frac{31}{64} a^{15} - \frac{25}{64} a^{13} + \frac{17}{64} a^{11} - \frac{9}{64} a^{9} + \frac{1}{64} a^{7} + \frac{7}{64} a^{5} - \frac{15}{64} a^{3} + \frac{49885262}{235418369} a$, $\frac{1}{60267102464} a^{24} + \frac{7}{60267102464} a^{22} - \frac{79}{256} a^{20} + \frac{87}{256} a^{18} - \frac{31}{256} a^{16} - \frac{89}{256} a^{14} + \frac{17}{256} a^{12} - \frac{9}{256} a^{10} + \frac{65}{256} a^{8} + \frac{71}{256} a^{6} + \frac{113}{256} a^{4} + \frac{24942631}{470836738} a^{2} - \frac{30409976}{235418369}$, $\frac{1}{241068409856} a^{25} + \frac{7}{241068409856} a^{23} - \frac{79}{1024} a^{21} - \frac{425}{1024} a^{19} - \frac{31}{1024} a^{17} - \frac{89}{1024} a^{15} + \frac{273}{1024} a^{13} - \frac{9}{1024} a^{11} - \frac{191}{1024} a^{9} - \frac{185}{1024} a^{7} - \frac{399}{1024} a^{5} + \frac{495779369}{1883346952} a^{3} - \frac{265828345}{941673476} a$, $\frac{1}{964273639424} a^{26} + \frac{7}{964273639424} a^{24} - \frac{79}{964273639424} a^{22} - \frac{681}{4096} a^{20} - \frac{799}{4096} a^{18} + \frac{1703}{4096} a^{16} + \frac{1553}{4096} a^{14} - \frac{265}{4096} a^{12} - \frac{1983}{4096} a^{10} + \frac{1607}{4096} a^{8} + \frac{881}{4096} a^{6} + \frac{24942631}{7533387808} a^{4} - \frac{3801247}{470836738} a^{2} + \frac{4634296}{235418369}$, $\frac{1}{3857094557696} a^{27} + \frac{7}{3857094557696} a^{25} - \frac{79}{3857094557696} a^{23} - \frac{681}{16384} a^{21} + \frac{3297}{16384} a^{19} - \frac{2393}{16384} a^{17} + \frac{1553}{16384} a^{15} + \frac{7927}{16384} a^{13} + \frac{2113}{16384} a^{11} + \frac{1607}{16384} a^{9} + \frac{881}{16384} a^{7} + \frac{7558330439}{30133551232} a^{5} - \frac{474637985}{1883346952} a^{3} + \frac{240052665}{941673476} a$, $\frac{1}{15428378230784} a^{28} + \frac{7}{15428378230784} a^{26} - \frac{79}{15428378230784} a^{24} + \frac{599}{15428378230784} a^{22} + \frac{7393}{65536} a^{20} + \frac{9895}{65536} a^{18} - \frac{10735}{65536} a^{16} + \frac{3831}{65536} a^{14} + \frac{6209}{65536} a^{12} + \frac{13895}{65536} a^{10} - \frac{27791}{65536} a^{8} + \frac{24942631}{120534204928} a^{6} - \frac{3801247}{7533387808} a^{4} + \frac{579287}{470836738} a^{2} - \frac{706168}{235418369}$, $\frac{1}{61713512923136} a^{29} + \frac{7}{61713512923136} a^{27} - \frac{79}{61713512923136} a^{25} + \frac{599}{61713512923136} a^{23} + \frac{7393}{262144} a^{21} + \frac{9895}{262144} a^{19} + \frac{54801}{262144} a^{17} - \frac{127241}{262144} a^{15} + \frac{6209}{262144} a^{13} - \frac{117177}{262144} a^{11} - \frac{93327}{262144} a^{9} - \frac{120509262297}{482136819712} a^{7} + \frac{7529586561}{30133551232} a^{5} - \frac{470257451}{1883346952} a^{3} + \frac{234712201}{941673476} a$, $\frac{1}{246854051692544} a^{30} + \frac{7}{246854051692544} a^{28} - \frac{79}{246854051692544} a^{26} + \frac{599}{246854051692544} a^{24} - \frac{4127}{246854051692544} a^{22} + \frac{337575}{1048576} a^{20} - \frac{10735}{1048576} a^{18} - \frac{61705}{1048576} a^{16} - \frac{321471}{1048576} a^{14} - \frac{313785}{1048576} a^{12} - \frac{421007}{1048576} a^{10} + \frac{24942631}{1928547278848} a^{8} - \frac{3801247}{120534204928} a^{6} + \frac{579287}{7533387808} a^{4} - \frac{88271}{470836738} a^{2} + \frac{107576}{235418369}$, $\frac{1}{987416206770176} a^{31} + \frac{7}{987416206770176} a^{29} - \frac{79}{987416206770176} a^{27} + \frac{599}{987416206770176} a^{25} - \frac{4127}{987416206770176} a^{23} + \frac{337575}{4194304} a^{21} + \frac{2086417}{4194304} a^{19} + \frac{986871}{4194304} a^{17} - \frac{321471}{4194304} a^{15} - \frac{313785}{4194304} a^{13} - \frac{421007}{4194304} a^{11} - \frac{1928522336217}{7714189115392} a^{9} + \frac{120530403681}{482136819712} a^{7} - \frac{7532808521}{30133551232} a^{5} + \frac{470748467}{1883346952} a^{3} - \frac{235310793}{941673476} a$, $\frac{1}{3949664827080704} a^{32} + \frac{7}{3949664827080704} a^{30} - \frac{79}{3949664827080704} a^{28} + \frac{599}{3949664827080704} a^{26} - \frac{4127}{3949664827080704} a^{24} + \frac{27559}{3949664827080704} a^{22} - \frac{3156463}{16777216} a^{20} + \frac{6229751}{16777216} a^{18} - \frac{5564351}{16777216} a^{16} + \frac{734791}{16777216} a^{14} - \frac{1469583}{16777216} a^{12} + \frac{24942631}{30856756461568} a^{10} - \frac{3801247}{1928547278848} a^{8} + \frac{579287}{120534204928} a^{6} - \frac{88271}{7533387808} a^{4} + \frac{13447}{470836738} a^{2} - \frac{16376}{235418369}$, $\frac{1}{15798659308322816} a^{33} + \frac{7}{15798659308322816} a^{31} - \frac{79}{15798659308322816} a^{29} + \frac{599}{15798659308322816} a^{27} - \frac{4127}{15798659308322816} a^{25} + \frac{27559}{15798659308322816} a^{23} - \frac{3156463}{67108864} a^{21} + \frac{6229751}{67108864} a^{19} - \frac{5564351}{67108864} a^{17} + \frac{17512007}{67108864} a^{15} - \frac{1469583}{67108864} a^{13} - \frac{61713487980505}{123427025846272} a^{11} + \frac{3857090756449}{7714189115392} a^{9} - \frac{241067830569}{482136819712} a^{7} + \frac{15066687345}{30133551232} a^{5} - \frac{941660029}{1883346952} a^{3} + \frac{235410181}{470836738} a$, $\frac{1}{63194637233291264} a^{34} + \frac{7}{63194637233291264} a^{32} - \frac{79}{63194637233291264} a^{30} + \frac{599}{63194637233291264} a^{28} - \frac{4127}{63194637233291264} a^{26} + \frac{27559}{63194637233291264} a^{24} - \frac{181999}{63194637233291264} a^{22} + \frac{106893047}{268435456} a^{20} - \frac{106227647}{268435456} a^{18} + \frac{51066439}{268435456} a^{16} - \frac{102132879}{268435456} a^{14} + \frac{24942631}{493708103385088} a^{12} - \frac{3801247}{30856756461568} a^{10} + \frac{579287}{1928547278848} a^{8} - \frac{88271}{120534204928} a^{6} + \frac{13447}{7533387808} a^{4} - \frac{2047}{470836738} a^{2} + \frac{2488}{235418369}$, $\frac{1}{252778548933165056} a^{35} + \frac{7}{252778548933165056} a^{33} - \frac{79}{252778548933165056} a^{31} + \frac{599}{252778548933165056} a^{29} - \frac{4127}{252778548933165056} a^{27} + \frac{27559}{252778548933165056} a^{25} - \frac{181999}{252778548933165056} a^{23} + \frac{106893047}{1073741824} a^{21} - \frac{374663103}{1073741824} a^{19} - \frac{485804473}{1073741824} a^{17} - \frac{370568335}{1073741824} a^{15} - \frac{493708078442457}{1974832413540352} a^{13} + \frac{30856752660321}{123427025846272} a^{11} - \frac{1928546699561}{7714189115392} a^{9} + \frac{120534116657}{482136819712} a^{7} - \frac{7533374361}{30133551232} a^{5} + \frac{470834691}{1883346952} a^{3} - \frac{235415881}{941673476} a$, $\frac{1}{1011114195732660224} a^{36} + \frac{7}{1011114195732660224} a^{34} - \frac{79}{1011114195732660224} a^{32} + \frac{599}{1011114195732660224} a^{30} - \frac{4127}{1011114195732660224} a^{28} + \frac{27559}{1011114195732660224} a^{26} - \frac{181999}{1011114195732660224} a^{24} + \frac{1197047}{1011114195732660224} a^{22} - \frac{1716840383}{4294967296} a^{20} + \frac{856372807}{4294967296} a^{18} - \frac{1712745615}{4294967296} a^{16} + \frac{24942631}{7899329654161408} a^{14} - \frac{3801247}{493708103385088} a^{12} + \frac{579287}{30856756461568} a^{10} - \frac{88271}{1928547278848} a^{8} + \frac{13447}{120534204928} a^{6} - \frac{2047}{7533387808} a^{4} + \frac{311}{470836738} a^{2} - \frac{376}{235418369}$, $\frac{1}{4044456782930640896} a^{37} + \frac{7}{4044456782930640896} a^{35} - \frac{79}{4044456782930640896} a^{33} + \frac{599}{4044456782930640896} a^{31} - \frac{4127}{4044456782930640896} a^{29} + \frac{27559}{4044456782930640896} a^{27} - \frac{181999}{4044456782930640896} a^{25} + \frac{1197047}{4044456782930640896} a^{23} - \frac{1716840383}{17179869184} a^{21} - \frac{7733561785}{17179869184} a^{19} - \frac{6007712911}{17179869184} a^{17} - \frac{7899329629218777}{31597318616645632} a^{15} + \frac{493708099583841}{1974832413540352} a^{13} - \frac{30856755882281}{123427025846272} a^{11} + \frac{1928547190577}{7714189115392} a^{9} - \frac{120534191481}{482136819712} a^{7} + \frac{7533385761}{30133551232} a^{5} - \frac{470836427}{1883346952} a^{3} + \frac{235417993}{941673476} a$, $\frac{1}{16177827131722563584} a^{38} + \frac{7}{16177827131722563584} a^{36} - \frac{79}{16177827131722563584} a^{34} + \frac{599}{16177827131722563584} a^{32} - \frac{4127}{16177827131722563584} a^{30} + \frac{27559}{16177827131722563584} a^{28} - \frac{181999}{16177827131722563584} a^{26} + \frac{1197047}{16177827131722563584} a^{24} - \frac{7861439}{16177827131722563584} a^{22} + \frac{13741274695}{68719476736} a^{20} - \frac{27482549391}{68719476736} a^{18} + \frac{24942631}{126389274466582528} a^{16} - \frac{3801247}{7899329654161408} a^{14} + \frac{579287}{493708103385088} a^{12} - \frac{88271}{30856756461568} a^{10} + \frac{13447}{1928547278848} a^{8} - \frac{2047}{120534204928} a^{6} + \frac{311}{7533387808} a^{4} - \frac{47}{470836738} a^{2} + \frac{56}{235418369}$, $\frac{1}{64711308526890254336} a^{39} + \frac{7}{64711308526890254336} a^{37} - \frac{79}{64711308526890254336} a^{35} + \frac{599}{64711308526890254336} a^{33} - \frac{4127}{64711308526890254336} a^{31} + \frac{27559}{64711308526890254336} a^{29} - \frac{181999}{64711308526890254336} a^{27} + \frac{1197047}{64711308526890254336} a^{25} - \frac{7861439}{64711308526890254336} a^{23} + \frac{13741274695}{274877906944} a^{21} - \frac{27482549391}{274877906944} a^{19} + \frac{24942631}{505557097866330112} a^{17} - \frac{3801247}{31597318616645632} a^{15} + \frac{579287}{1974832413540352} a^{13} - \frac{88271}{123427025846272} a^{11} + \frac{13447}{7714189115392} a^{9} - \frac{2047}{482136819712} a^{7} + \frac{311}{30133551232} a^{5} - \frac{47}{1883346952} a^{3} + \frac{14}{235418369} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{29}{241068409856} a^{27} + \frac{25963647845}{241068409856} a^{5} \) (order $44$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{187}) \), \(\Q(\sqrt{-187}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(i, \sqrt{17})\), \(\Q(i, \sqrt{187})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{11}, \sqrt{17})\), \(\Q(\sqrt{-11}, \sqrt{17})\), \(\Q(\sqrt{-11}, \sqrt{-17})\), \(\Q(\sqrt{11}, \sqrt{-17})\), \(\Q(\zeta_{11})^+\), 8.0.313044726016.11, 10.0.219503494144.1, 10.10.304358957700017.1, 10.0.311663572684817408.3, 10.10.3428299299532991488.1, 10.0.3347948534700187.1, \(\Q(\zeta_{44})^+\), \(\Q(\zeta_{11})\), 20.0.97134182538664463559097634299838464.1, 20.0.11753236087178400090650813750280454144.4, \(\Q(\zeta_{44})\), 20.20.11753236087178400090650813750280454144.1, 20.0.11208759391001129236841977834969.1, 20.0.11753236087178400090650813750280454144.5, 20.0.11753236087178400090650813750280454144.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
11Data not computed
17Data not computed