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Properties

Label 40.0.138...736.1
Degree $40$
Signature $[0, 20]$
Discriminant $1.381\times 10^{74}$
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: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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(y^40 - 9*y^38 + 65*y^36 - 441*y^34 + 2929*y^32 - 19305*y^30 + 126881*y^28 - 833049*y^26 + 5467345*y^24 - 35877321*y^22 + 235418369*y^20 - 574037136*y^18 + 1399640320*y^16 - 3412168704*y^14 + 8315273216*y^12 - 20242759680*y^10 + 49140465664*y^8 - 118380036096*y^6 + 279172874240*y^4 - 618475290624*y^2 + 1099511627776, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{40} - 9 x^{38} + 65 x^{36} - 441 x^{34} + 2929 x^{32} - 19305 x^{30} + 126881 x^{28} + \cdots + 1099511627776 \) Copy content Toggle raw display

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

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(138138558520952628335754257563768298595689088682615055246398014526886772736\) \(\medspace = 2^{40}\cdot 11^{36}\cdot 17^{20}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(71.37\)
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:  $2\cdot 11^{9/10}17^{1/2}\approx 71.3687142898531$
Ramified primes:   \(2\), \(11\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{4}a^{21}-\frac{1}{4}a^{19}+\frac{1}{4}a^{17}-\frac{1}{4}a^{15}+\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{3766693904}a^{22}+\frac{7}{16}a^{20}+\frac{1}{16}a^{18}+\frac{7}{16}a^{16}+\frac{1}{16}a^{14}+\frac{7}{16}a^{12}+\frac{1}{16}a^{10}+\frac{7}{16}a^{8}+\frac{1}{16}a^{6}+\frac{7}{16}a^{4}+\frac{1}{16}a^{2}-\frac{35877321}{235418369}$, $\frac{1}{15066775616}a^{23}+\frac{7}{64}a^{21}-\frac{15}{64}a^{19}+\frac{23}{64}a^{17}-\frac{31}{64}a^{15}-\frac{25}{64}a^{13}+\frac{17}{64}a^{11}-\frac{9}{64}a^{9}+\frac{1}{64}a^{7}+\frac{7}{64}a^{5}-\frac{15}{64}a^{3}+\frac{49885262}{235418369}a$, $\frac{1}{60267102464}a^{24}+\frac{7}{60267102464}a^{22}-\frac{79}{256}a^{20}+\frac{87}{256}a^{18}-\frac{31}{256}a^{16}-\frac{89}{256}a^{14}+\frac{17}{256}a^{12}-\frac{9}{256}a^{10}+\frac{65}{256}a^{8}+\frac{71}{256}a^{6}+\frac{113}{256}a^{4}+\frac{24942631}{470836738}a^{2}-\frac{30409976}{235418369}$, $\frac{1}{241068409856}a^{25}+\frac{7}{241068409856}a^{23}-\frac{79}{1024}a^{21}-\frac{425}{1024}a^{19}-\frac{31}{1024}a^{17}-\frac{89}{1024}a^{15}+\frac{273}{1024}a^{13}-\frac{9}{1024}a^{11}-\frac{191}{1024}a^{9}-\frac{185}{1024}a^{7}-\frac{399}{1024}a^{5}+\frac{495779369}{1883346952}a^{3}-\frac{265828345}{941673476}a$, $\frac{1}{964273639424}a^{26}+\frac{7}{964273639424}a^{24}-\frac{79}{964273639424}a^{22}-\frac{681}{4096}a^{20}-\frac{799}{4096}a^{18}+\frac{1703}{4096}a^{16}+\frac{1553}{4096}a^{14}-\frac{265}{4096}a^{12}-\frac{1983}{4096}a^{10}+\frac{1607}{4096}a^{8}+\frac{881}{4096}a^{6}+\frac{24942631}{7533387808}a^{4}-\frac{3801247}{470836738}a^{2}+\frac{4634296}{235418369}$, $\frac{1}{3857094557696}a^{27}+\frac{7}{3857094557696}a^{25}-\frac{79}{3857094557696}a^{23}-\frac{681}{16384}a^{21}+\frac{3297}{16384}a^{19}-\frac{2393}{16384}a^{17}+\frac{1553}{16384}a^{15}+\frac{7927}{16384}a^{13}+\frac{2113}{16384}a^{11}+\frac{1607}{16384}a^{9}+\frac{881}{16384}a^{7}+\frac{7558330439}{30133551232}a^{5}-\frac{474637985}{1883346952}a^{3}+\frac{240052665}{941673476}a$, $\frac{1}{15428378230784}a^{28}+\frac{7}{15428378230784}a^{26}-\frac{79}{15428378230784}a^{24}+\frac{599}{15428378230784}a^{22}+\frac{7393}{65536}a^{20}+\frac{9895}{65536}a^{18}-\frac{10735}{65536}a^{16}+\frac{3831}{65536}a^{14}+\frac{6209}{65536}a^{12}+\frac{13895}{65536}a^{10}-\frac{27791}{65536}a^{8}+\frac{24942631}{120534204928}a^{6}-\frac{3801247}{7533387808}a^{4}+\frac{579287}{470836738}a^{2}-\frac{706168}{235418369}$, $\frac{1}{61713512923136}a^{29}+\frac{7}{61713512923136}a^{27}-\frac{79}{61713512923136}a^{25}+\frac{599}{61713512923136}a^{23}+\frac{7393}{262144}a^{21}+\frac{9895}{262144}a^{19}+\frac{54801}{262144}a^{17}-\frac{127241}{262144}a^{15}+\frac{6209}{262144}a^{13}-\frac{117177}{262144}a^{11}-\frac{93327}{262144}a^{9}-\frac{120509262297}{482136819712}a^{7}+\frac{7529586561}{30133551232}a^{5}-\frac{470257451}{1883346952}a^{3}+\frac{234712201}{941673476}a$, $\frac{1}{246854051692544}a^{30}+\frac{7}{246854051692544}a^{28}-\frac{79}{246854051692544}a^{26}+\frac{599}{246854051692544}a^{24}-\frac{4127}{246854051692544}a^{22}+\frac{337575}{1048576}a^{20}-\frac{10735}{1048576}a^{18}-\frac{61705}{1048576}a^{16}-\frac{321471}{1048576}a^{14}-\frac{313785}{1048576}a^{12}-\frac{421007}{1048576}a^{10}+\frac{24942631}{1928547278848}a^{8}-\frac{3801247}{120534204928}a^{6}+\frac{579287}{7533387808}a^{4}-\frac{88271}{470836738}a^{2}+\frac{107576}{235418369}$, $\frac{1}{987416206770176}a^{31}+\frac{7}{987416206770176}a^{29}-\frac{79}{987416206770176}a^{27}+\frac{599}{987416206770176}a^{25}-\frac{4127}{987416206770176}a^{23}+\frac{337575}{4194304}a^{21}+\frac{2086417}{4194304}a^{19}+\frac{986871}{4194304}a^{17}-\frac{321471}{4194304}a^{15}-\frac{313785}{4194304}a^{13}-\frac{421007}{4194304}a^{11}-\frac{1928522336217}{7714189115392}a^{9}+\frac{120530403681}{482136819712}a^{7}-\frac{7532808521}{30133551232}a^{5}+\frac{470748467}{1883346952}a^{3}-\frac{235310793}{941673476}a$, $\frac{1}{39\!\cdots\!04}a^{32}+\frac{7}{39\!\cdots\!04}a^{30}-\frac{79}{39\!\cdots\!04}a^{28}+\frac{599}{39\!\cdots\!04}a^{26}-\frac{4127}{39\!\cdots\!04}a^{24}+\frac{27559}{39\!\cdots\!04}a^{22}-\frac{3156463}{16777216}a^{20}+\frac{6229751}{16777216}a^{18}-\frac{5564351}{16777216}a^{16}+\frac{734791}{16777216}a^{14}-\frac{1469583}{16777216}a^{12}+\frac{24942631}{30856756461568}a^{10}-\frac{3801247}{1928547278848}a^{8}+\frac{579287}{120534204928}a^{6}-\frac{88271}{7533387808}a^{4}+\frac{13447}{470836738}a^{2}-\frac{16376}{235418369}$, $\frac{1}{15\!\cdots\!16}a^{33}+\frac{7}{15\!\cdots\!16}a^{31}-\frac{79}{15\!\cdots\!16}a^{29}+\frac{599}{15\!\cdots\!16}a^{27}-\frac{4127}{15\!\cdots\!16}a^{25}+\frac{27559}{15\!\cdots\!16}a^{23}-\frac{3156463}{67108864}a^{21}+\frac{6229751}{67108864}a^{19}-\frac{5564351}{67108864}a^{17}+\frac{17512007}{67108864}a^{15}-\frac{1469583}{67108864}a^{13}-\frac{61713487980505}{123427025846272}a^{11}+\frac{3857090756449}{7714189115392}a^{9}-\frac{241067830569}{482136819712}a^{7}+\frac{15066687345}{30133551232}a^{5}-\frac{941660029}{1883346952}a^{3}+\frac{235410181}{470836738}a$, $\frac{1}{63\!\cdots\!64}a^{34}+\frac{7}{63\!\cdots\!64}a^{32}-\frac{79}{63\!\cdots\!64}a^{30}+\frac{599}{63\!\cdots\!64}a^{28}-\frac{4127}{63\!\cdots\!64}a^{26}+\frac{27559}{63\!\cdots\!64}a^{24}-\frac{181999}{63\!\cdots\!64}a^{22}+\frac{106893047}{268435456}a^{20}-\frac{106227647}{268435456}a^{18}+\frac{51066439}{268435456}a^{16}-\frac{102132879}{268435456}a^{14}+\frac{24942631}{493708103385088}a^{12}-\frac{3801247}{30856756461568}a^{10}+\frac{579287}{1928547278848}a^{8}-\frac{88271}{120534204928}a^{6}+\frac{13447}{7533387808}a^{4}-\frac{2047}{470836738}a^{2}+\frac{2488}{235418369}$, $\frac{1}{25\!\cdots\!56}a^{35}+\frac{7}{25\!\cdots\!56}a^{33}-\frac{79}{25\!\cdots\!56}a^{31}+\frac{599}{25\!\cdots\!56}a^{29}-\frac{4127}{25\!\cdots\!56}a^{27}+\frac{27559}{25\!\cdots\!56}a^{25}-\frac{181999}{25\!\cdots\!56}a^{23}+\frac{106893047}{1073741824}a^{21}-\frac{374663103}{1073741824}a^{19}-\frac{485804473}{1073741824}a^{17}-\frac{370568335}{1073741824}a^{15}-\frac{493708078442457}{19\!\cdots\!52}a^{13}+\frac{30856752660321}{123427025846272}a^{11}-\frac{1928546699561}{7714189115392}a^{9}+\frac{120534116657}{482136819712}a^{7}-\frac{7533374361}{30133551232}a^{5}+\frac{470834691}{1883346952}a^{3}-\frac{235415881}{941673476}a$, $\frac{1}{10\!\cdots\!24}a^{36}+\frac{7}{10\!\cdots\!24}a^{34}-\frac{79}{10\!\cdots\!24}a^{32}+\frac{599}{10\!\cdots\!24}a^{30}-\frac{4127}{10\!\cdots\!24}a^{28}+\frac{27559}{10\!\cdots\!24}a^{26}-\frac{181999}{10\!\cdots\!24}a^{24}+\frac{1197047}{10\!\cdots\!24}a^{22}-\frac{1716840383}{4294967296}a^{20}+\frac{856372807}{4294967296}a^{18}-\frac{1712745615}{4294967296}a^{16}+\frac{24942631}{78\!\cdots\!08}a^{14}-\frac{3801247}{493708103385088}a^{12}+\frac{579287}{30856756461568}a^{10}-\frac{88271}{1928547278848}a^{8}+\frac{13447}{120534204928}a^{6}-\frac{2047}{7533387808}a^{4}+\frac{311}{470836738}a^{2}-\frac{376}{235418369}$, $\frac{1}{40\!\cdots\!96}a^{37}+\frac{7}{40\!\cdots\!96}a^{35}-\frac{79}{40\!\cdots\!96}a^{33}+\frac{599}{40\!\cdots\!96}a^{31}-\frac{4127}{40\!\cdots\!96}a^{29}+\frac{27559}{40\!\cdots\!96}a^{27}-\frac{181999}{40\!\cdots\!96}a^{25}+\frac{1197047}{40\!\cdots\!96}a^{23}-\frac{1716840383}{17179869184}a^{21}-\frac{7733561785}{17179869184}a^{19}-\frac{6007712911}{17179869184}a^{17}-\frac{78\!\cdots\!77}{31\!\cdots\!32}a^{15}+\frac{493708099583841}{19\!\cdots\!52}a^{13}-\frac{30856755882281}{123427025846272}a^{11}+\frac{1928547190577}{7714189115392}a^{9}-\frac{120534191481}{482136819712}a^{7}+\frac{7533385761}{30133551232}a^{5}-\frac{470836427}{1883346952}a^{3}+\frac{235417993}{941673476}a$, $\frac{1}{16\!\cdots\!84}a^{38}+\frac{7}{16\!\cdots\!84}a^{36}-\frac{79}{16\!\cdots\!84}a^{34}+\frac{599}{16\!\cdots\!84}a^{32}-\frac{4127}{16\!\cdots\!84}a^{30}+\frac{27559}{16\!\cdots\!84}a^{28}-\frac{181999}{16\!\cdots\!84}a^{26}+\frac{1197047}{16\!\cdots\!84}a^{24}-\frac{7861439}{16\!\cdots\!84}a^{22}+\frac{13741274695}{68719476736}a^{20}-\frac{27482549391}{68719476736}a^{18}+\frac{24942631}{12\!\cdots\!28}a^{16}-\frac{3801247}{78\!\cdots\!08}a^{14}+\frac{579287}{493708103385088}a^{12}-\frac{88271}{30856756461568}a^{10}+\frac{13447}{1928547278848}a^{8}-\frac{2047}{120534204928}a^{6}+\frac{311}{7533387808}a^{4}-\frac{47}{470836738}a^{2}+\frac{56}{235418369}$, $\frac{1}{64\!\cdots\!36}a^{39}+\frac{7}{64\!\cdots\!36}a^{37}-\frac{79}{64\!\cdots\!36}a^{35}+\frac{599}{64\!\cdots\!36}a^{33}-\frac{4127}{64\!\cdots\!36}a^{31}+\frac{27559}{64\!\cdots\!36}a^{29}-\frac{181999}{64\!\cdots\!36}a^{27}+\frac{1197047}{64\!\cdots\!36}a^{25}-\frac{7861439}{64\!\cdots\!36}a^{23}+\frac{13741274695}{274877906944}a^{21}-\frac{27482549391}{274877906944}a^{19}+\frac{24942631}{50\!\cdots\!12}a^{17}-\frac{3801247}{31\!\cdots\!32}a^{15}+\frac{579287}{19\!\cdots\!52}a^{13}-\frac{88271}{123427025846272}a^{11}+\frac{13447}{7714189115392}a^{9}-\frac{2047}{482136819712}a^{7}+\frac{311}{30133551232}a^{5}-\frac{47}{1883346952}a^{3}+\frac{14}{235418369}a$ Copy content Toggle raw display

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

not computed

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:   \( \frac{29}{241068409856} a^{27} + \frac{25963647845}{241068409856} a^{5} \)  (order $44$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
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^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);
 
[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^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);
 
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^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);
 
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

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

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)
 
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.

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]
 

Frobenius cycle types

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

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

# 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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $20$$2$$10$$20$
Deg $20$$2$$10$$20$
\(11\) Copy content Toggle raw display 11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
\(17\) Copy content Toggle raw display 17.20.10.1$x^{20} + 1530 x^{19} + 1053575 x^{18} + 430023330 x^{17} + 115219345095 x^{16} + 21179127107942 x^{15} + 2705561983266777 x^{14} + 237314944467250907 x^{13} + 13696435412906736123 x^{12} + 471506205668928373499 x^{11} + 7493452481890385447638 x^{10} + 8015640618238448390772 x^{9} + 3961351679336933294924 x^{8} + 1343700033572449112637 x^{7} + 6339565094661515050803 x^{6} + 96914313552023485021024 x^{5} + 96698524475936930004680 x^{4} + 70077298249497512241355 x^{3} + 85352967358824559052487 x^{2} + 101334540265351382936671 x + 31078993826931964273351$$2$$10$$10$20T3$[\ ]_{2}^{10}$
17.20.10.1$x^{20} + 1530 x^{19} + 1053575 x^{18} + 430023330 x^{17} + 115219345095 x^{16} + 21179127107942 x^{15} + 2705561983266777 x^{14} + 237314944467250907 x^{13} + 13696435412906736123 x^{12} + 471506205668928373499 x^{11} + 7493452481890385447638 x^{10} + 8015640618238448390772 x^{9} + 3961351679336933294924 x^{8} + 1343700033572449112637 x^{7} + 6339565094661515050803 x^{6} + 96914313552023485021024 x^{5} + 96698524475936930004680 x^{4} + 70077298249497512241355 x^{3} + 85352967358824559052487 x^{2} + 101334540265351382936671 x + 31078993826931964273351$$2$$10$$10$20T3$[\ ]_{2}^{10}$