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Properties

Label 40.0.280...496.1
Degree $40$
Signature $[0, 20]$
Discriminant $2.807\times 10^{69}$
Root discriminant \(54.48\)
Ramified primes $2,11$
Class number $2605$ (GRH)
Class group [2605] (GRH)
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 257*x^32 + 14016*x^24 + 105419*x^16 + 23219*x^8 + 1)
 
gp: K = bnfinit(y^40 + 257*y^32 + 14016*y^24 + 105419*y^16 + 23219*y^8 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 257*x^32 + 14016*x^24 + 105419*x^16 + 23219*x^8 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 257*x^32 + 14016*x^24 + 105419*x^16 + 23219*x^8 + 1)
 

\( x^{40} + 257x^{32} + 14016x^{24} + 105419x^{16} + 23219x^{8} + 1 \) 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:   \(2806502314667513790456400791754773417894971971642777758942872517738496\) \(\medspace = 2^{120}\cdot 11^{32}\) 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:  \(54.48\)
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:  $2^{3}11^{4/5}\approx 54.47586502017841$
Ramified primes:   \(2\), \(11\) 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:  \(176=2^{4}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(3,·)$, $\chi_{176}(133,·)$, $\chi_{176}(135,·)$, $\chi_{176}(9,·)$, $\chi_{176}(141,·)$, $\chi_{176}(15,·)$, $\chi_{176}(147,·)$, $\chi_{176}(23,·)$, $\chi_{176}(25,·)$, $\chi_{176}(155,·)$, $\chi_{176}(157,·)$, $\chi_{176}(5,·)$, $\chi_{176}(27,·)$, $\chi_{176}(37,·)$, $\chi_{176}(31,·)$, $\chi_{176}(169,·)$, $\chi_{176}(45,·)$, $\chi_{176}(47,·)$, $\chi_{176}(49,·)$, $\chi_{176}(53,·)$, $\chi_{176}(137,·)$, $\chi_{176}(159,·)$, $\chi_{176}(67,·)$, $\chi_{176}(69,·)$, $\chi_{176}(71,·)$, $\chi_{176}(75,·)$, $\chi_{176}(81,·)$, $\chi_{176}(163,·)$, $\chi_{176}(89,·)$, $\chi_{176}(91,·)$, $\chi_{176}(93,·)$, $\chi_{176}(97,·)$, $\chi_{176}(59,·)$, $\chi_{176}(103,·)$, $\chi_{176}(111,·)$, $\chi_{176}(113,·)$, $\chi_{176}(115,·)$, $\chi_{176}(119,·)$, $\chi_{176}(125,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{22148330043533}a^{32}+\frac{2726870929642}{22148330043533}a^{24}-\frac{7580340779727}{22148330043533}a^{16}+\frac{6188800161857}{22148330043533}a^{8}+\frac{6246971138049}{22148330043533}$, $\frac{1}{22148330043533}a^{33}+\frac{2726870929642}{22148330043533}a^{25}-\frac{7580340779727}{22148330043533}a^{17}+\frac{6188800161857}{22148330043533}a^{9}+\frac{6246971138049}{22148330043533}a$, $\frac{1}{22148330043533}a^{34}+\frac{2726870929642}{22148330043533}a^{26}-\frac{7580340779727}{22148330043533}a^{18}+\frac{6188800161857}{22148330043533}a^{10}+\frac{6246971138049}{22148330043533}a^{2}$, $\frac{1}{22148330043533}a^{35}+\frac{2726870929642}{22148330043533}a^{27}-\frac{7580340779727}{22148330043533}a^{19}+\frac{6188800161857}{22148330043533}a^{11}+\frac{6246971138049}{22148330043533}a^{3}$, $\frac{1}{22148330043533}a^{36}+\frac{2726870929642}{22148330043533}a^{28}-\frac{7580340779727}{22148330043533}a^{20}+\frac{6188800161857}{22148330043533}a^{12}+\frac{6246971138049}{22148330043533}a^{4}$, $\frac{1}{22148330043533}a^{37}+\frac{2726870929642}{22148330043533}a^{29}-\frac{7580340779727}{22148330043533}a^{21}+\frac{6188800161857}{22148330043533}a^{13}+\frac{6246971138049}{22148330043533}a^{5}$, $\frac{1}{22148330043533}a^{38}+\frac{2726870929642}{22148330043533}a^{30}-\frac{7580340779727}{22148330043533}a^{22}+\frac{6188800161857}{22148330043533}a^{14}+\frac{6246971138049}{22148330043533}a^{6}$, $\frac{1}{22148330043533}a^{39}+\frac{2726870929642}{22148330043533}a^{31}-\frac{7580340779727}{22148330043533}a^{23}+\frac{6188800161857}{22148330043533}a^{15}+\frac{6246971138049}{22148330043533}a^{7}$ 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

$C_{2605}$, which has order $2605$ (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:   \( \frac{43114401572}{22148330043533} a^{35} + \frac{11079877784514}{22148330043533} a^{27} + \frac{604158913402484}{22148330043533} a^{19} + \frac{4538168045437461}{22148330043533} a^{11} + \frac{960702298577351}{22148330043533} a^{3} \)  (order $16$) 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:   $\frac{2378192679}{22148330043533}a^{33}+\frac{611758976686}{22148330043533}a^{25}+\frac{33479942029207}{22148330043533}a^{17}+\frac{258956167086371}{22148330043533}a^{9}+\frac{101926540832058}{22148330043533}a$, $a$, $\frac{1496826812680}{22148330043533}a^{38}+\frac{384683031892851}{22148330043533}a^{30}+\frac{20\!\cdots\!32}{22148330043533}a^{22}+\frac{15\!\cdots\!67}{22148330043533}a^{14}+\frac{34\!\cdots\!26}{22148330043533}a^{6}$, $\frac{58742050677}{22148330043533}a^{35}+\frac{15096876188376}{22148330043533}a^{27}+\frac{823368847411579}{22148330043533}a^{19}+\frac{61\!\cdots\!53}{22148330043533}a^{11}+\frac{13\!\cdots\!83}{22148330043533}a^{3}$, $\frac{1791689491944}{22148330043533}a^{38}-\frac{10370405466}{22148330043533}a^{34}+\frac{460464750524328}{22148330043533}a^{30}-\frac{2665377088768}{22148330043533}a^{26}+\frac{25\!\cdots\!77}{22148330043533}a^{22}-\frac{145398998848981}{22148330043533}a^{18}+\frac{18\!\cdots\!86}{22148330043533}a^{14}-\frac{10\!\cdots\!69}{22148330043533}a^{10}+\frac{41\!\cdots\!81}{22148330043533}a^{6}-\frac{273435950860925}{22148330043533}a^{2}-1$, $\frac{1791689491944}{22148330043533}a^{38}+\frac{513313738084}{22148330043533}a^{37}+\frac{1594541}{246385481}a^{36}+\frac{460464750524328}{22148330043533}a^{30}+\frac{131920969892455}{22148330043533}a^{29}+\frac{409801234}{246385481}a^{28}+\frac{25\!\cdots\!77}{22148330043533}a^{22}+\frac{71\!\cdots\!72}{22148330043533}a^{21}+\frac{22350169159}{246385481}a^{20}+\frac{18\!\cdots\!86}{22148330043533}a^{14}+\frac{54\!\cdots\!76}{22148330043533}a^{13}+\frac{168154800730}{246385481}a^{12}+\frac{41\!\cdots\!81}{22148330043533}a^{6}+\frac{11\!\cdots\!59}{22148330043533}a^{5}+\frac{37547254457}{246385481}a^{4}$, $\frac{1791689491944}{22148330043533}a^{38}-\frac{43114401572}{22148330043533}a^{35}+\frac{460464750524328}{22148330043533}a^{30}-\frac{11079877784514}{22148330043533}a^{27}+\frac{25\!\cdots\!77}{22148330043533}a^{22}-\frac{604158913402484}{22148330043533}a^{19}+\frac{18\!\cdots\!86}{22148330043533}a^{14}-\frac{45\!\cdots\!61}{22148330043533}a^{11}+\frac{41\!\cdots\!81}{22148330043533}a^{6}-\frac{960702298577351}{22148330043533}a^{3}+1$, $\frac{6308913397526}{22148330043533}a^{39}-\frac{43114401572}{22148330043533}a^{35}+\frac{16\!\cdots\!55}{22148330043533}a^{31}-\frac{11079877784514}{22148330043533}a^{27}+\frac{88\!\cdots\!96}{22148330043533}a^{23}-\frac{604158913402484}{22148330043533}a^{19}+\frac{66\!\cdots\!34}{22148330043533}a^{15}-\frac{45\!\cdots\!61}{22148330043533}a^{11}+\frac{14\!\cdots\!90}{22148330043533}a^{7}-\frac{960702298577351}{22148330043533}a^{3}-a$, $\frac{1594541}{246385481}a^{36}-\frac{20189716212}{22148330043533}a^{34}+\frac{409801234}{246385481}a^{28}-\frac{5188940493763}{22148330043533}a^{26}+\frac{22350169159}{246385481}a^{20}-\frac{283025395871412}{22148330043533}a^{18}+\frac{168154800730}{246385481}a^{12}-\frac{21\!\cdots\!93}{22148330043533}a^{10}+\frac{37547254457}{246385481}a^{4}-\frac{460070271039662}{22148330043533}a^{2}+1$, $\frac{5231730401719}{22148330043533}a^{39}-\frac{513313738084}{22148330043533}a^{37}+\frac{4562067107}{22148330043533}a^{33}+\frac{13\!\cdots\!22}{22148330043533}a^{31}-\frac{131920969892455}{22148330043533}a^{29}+\frac{1171942089901}{22148330043533}a^{25}+\frac{73\!\cdots\!44}{22148330043533}a^{23}-\frac{71\!\cdots\!72}{22148330043533}a^{21}+\frac{63815461862317}{22148330043533}a^{17}+\frac{55\!\cdots\!68}{22148330043533}a^{15}-\frac{54\!\cdots\!76}{22148330043533}a^{13}+\frac{474674153821901}{22148330043533}a^{9}+\frac{12\!\cdots\!42}{22148330043533}a^{7}-\frac{11\!\cdots\!59}{22148330043533}a^{5}+\frac{77834996691430}{22148330043533}a$, $\frac{6308913397526}{22148330043533}a^{39}-\frac{43114401572}{22148330043533}a^{35}+\frac{2378192679}{22148330043533}a^{33}+\frac{16\!\cdots\!55}{22148330043533}a^{31}-\frac{11079877784514}{22148330043533}a^{27}+\frac{611758976686}{22148330043533}a^{25}+\frac{88\!\cdots\!96}{22148330043533}a^{23}-\frac{604158913402484}{22148330043533}a^{19}+\frac{33479942029207}{22148330043533}a^{17}+\frac{66\!\cdots\!34}{22148330043533}a^{15}-\frac{45\!\cdots\!61}{22148330043533}a^{11}+\frac{258956167086371}{22148330043533}a^{9}+\frac{14\!\cdots\!90}{22148330043533}a^{7}-\frac{960702298577351}{22148330043533}a^{3}+\frac{101926540832058}{22148330043533}a$, $\frac{1791689491944}{22148330043533}a^{38}+\frac{10370405466}{22148330043533}a^{35}+\frac{460464750524328}{22148330043533}a^{30}+\frac{2665377088768}{22148330043533}a^{27}+\frac{25\!\cdots\!77}{22148330043533}a^{22}+\frac{145398998848981}{22148330043533}a^{19}+\frac{18\!\cdots\!86}{22148330043533}a^{14}+\frac{10\!\cdots\!69}{22148330043533}a^{11}+\frac{41\!\cdots\!81}{22148330043533}a^{6}+\frac{273435950860925}{22148330043533}a^{3}+1$, $\frac{20189716212}{22148330043533}a^{34}-\frac{2734969148}{22148330043533}a^{33}+\frac{1827097959}{22148330043533}a^{32}+\frac{5188940493763}{22148330043533}a^{26}-\frac{701996796988}{22148330043533}a^{25}+\frac{469945292913}{22148330043533}a^{24}+\frac{283025395871412}{22148330043533}a^{18}-\frac{38108121659660}{22148330043533}a^{17}+\frac{25707340202657}{22148330043533}a^{16}+\frac{21\!\cdots\!93}{22148330043533}a^{10}-\frac{276922940567475}{22148330043533}a^{9}+\frac{197751213254426}{22148330043533}a^{8}+\frac{460070271039662}{22148330043533}a^{2}-\frac{40561756498027}{22148330043533}a+\frac{37273240193403}{22148330043533}$, $\frac{1791689491944}{22148330043533}a^{38}+\frac{664838343235}{22148330043533}a^{37}+\frac{1594541}{246385481}a^{36}+\frac{460464750524328}{22148330043533}a^{30}+\frac{170864426195970}{22148330043533}a^{29}+\frac{409801234}{246385481}a^{28}+\frac{25\!\cdots\!77}{22148330043533}a^{22}+\frac{93\!\cdots\!30}{22148330043533}a^{21}+\frac{22350169159}{246385481}a^{20}+\frac{18\!\cdots\!86}{22148330043533}a^{14}+\frac{70\!\cdots\!05}{22148330043533}a^{13}+\frac{168154800730}{246385481}a^{12}+\frac{41\!\cdots\!81}{22148330043533}a^{6}+\frac{15\!\cdots\!13}{22148330043533}a^{5}+\frac{37547254457}{246385481}a^{4}$, $\frac{16916599641814}{22148330043533}a^{39}-\frac{9819310746}{22148330043533}a^{33}+\frac{551094720}{22148330043533}a^{32}+\frac{43\!\cdots\!59}{22148330043533}a^{31}-\frac{2523563404995}{22148330043533}a^{25}+\frac{141813683773}{22148330043533}a^{24}+\frac{23\!\cdots\!84}{22148330043533}a^{23}-\frac{137626397022431}{22148330043533}a^{17}+\frac{7772601826550}{22148330043533}a^{16}+\frac{17\!\cdots\!59}{22148330043533}a^{15}-\frac{10\!\cdots\!24}{22148330043533}a^{9}+\frac{61204953831945}{22148330043533}a^{8}+\frac{39\!\cdots\!85}{22148330043533}a^{7}-\frac{208782650222270}{22148330043533}a+\frac{42504970595122}{22148330043533}$, $\frac{6308913397526}{22148330043533}a^{39}+\frac{1496826812680}{22148330043533}a^{38}+\frac{513313738084}{22148330043533}a^{37}+\frac{16\!\cdots\!55}{22148330043533}a^{31}+\frac{384683031892851}{22148330043533}a^{30}+\frac{131920969892455}{22148330043533}a^{29}+\frac{88\!\cdots\!96}{22148330043533}a^{23}+\frac{20\!\cdots\!32}{22148330043533}a^{22}+\frac{71\!\cdots\!72}{22148330043533}a^{21}+\frac{66\!\cdots\!34}{22148330043533}a^{15}+\frac{15\!\cdots\!67}{22148330043533}a^{14}+\frac{54\!\cdots\!76}{22148330043533}a^{13}+\frac{14\!\cdots\!90}{22148330043533}a^{7}+\frac{34\!\cdots\!26}{22148330043533}a^{6}+\frac{11\!\cdots\!59}{22148330043533}a^{5}$, $\frac{10607686244288}{22148330043533}a^{39}-\frac{858731936987}{22148330043533}a^{37}-\frac{1594541}{246385481}a^{36}+\frac{22924685360}{22148330043533}a^{34}+\frac{27\!\cdots\!04}{22148330043533}a^{31}-\frac{220694370736755}{22148330043533}a^{29}-\frac{409801234}{246385481}a^{28}+\frac{5890937290751}{22148330043533}a^{26}+\frac{14\!\cdots\!88}{22148330043533}a^{23}-\frac{12\!\cdots\!25}{22148330043533}a^{21}-\frac{22350169159}{246385481}a^{20}+\frac{321133517531072}{22148330043533}a^{18}+\frac{11\!\cdots\!25}{22148330043533}a^{15}-\frac{90\!\cdots\!09}{22148330043533}a^{13}-\frac{168154800730}{246385481}a^{12}+\frac{24\!\cdots\!68}{22148330043533}a^{10}+\frac{24\!\cdots\!95}{22148330043533}a^{7}-\frac{19\!\cdots\!44}{22148330043533}a^{5}-\frac{37547254457}{246385481}a^{4}+\frac{500632027537689}{22148330043533}a^{2}-1$, $\frac{1791689491944}{22148330043533}a^{38}+\frac{31111216398}{22148330043533}a^{35}+\frac{460464750524328}{22148330043533}a^{30}+\frac{7996131266304}{22148330043533}a^{27}+\frac{25\!\cdots\!77}{22148330043533}a^{22}+\frac{436196996546943}{22148330043533}a^{19}+\frac{18\!\cdots\!86}{22148330043533}a^{14}+\frac{32\!\cdots\!07}{22148330043533}a^{11}+\frac{41\!\cdots\!81}{22148330043533}a^{6}+\frac{798159522539242}{22148330043533}a^{3}+1$, $\frac{513313738084}{22148330043533}a^{37}-\frac{43114401572}{22148330043533}a^{35}-\frac{551094720}{22148330043533}a^{32}+\frac{131920969892455}{22148330043533}a^{29}-\frac{11079877784514}{22148330043533}a^{27}-\frac{141813683773}{22148330043533}a^{24}+\frac{71\!\cdots\!72}{22148330043533}a^{21}-\frac{604158913402484}{22148330043533}a^{19}-\frac{7772601826550}{22148330043533}a^{16}+\frac{54\!\cdots\!76}{22148330043533}a^{13}-\frac{45\!\cdots\!61}{22148330043533}a^{11}-\frac{61204953831945}{22148330043533}a^{8}+\frac{11\!\cdots\!59}{22148330043533}a^{5}-\frac{960702298577351}{22148330043533}a^{3}-\frac{42504970595122}{22148330043533}$ Copy content Toggle raw display (assuming GRH)
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:  \( 4506035972876552.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 4506035972876552.0 \cdot 2605}{16\cdot\sqrt{2806502314667513790456400791754773417894971971642777758942872517738496}}\cr\approx \mathstrut & 0.127349447243315 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^40 + 257*x^32 + 14016*x^24 + 105419*x^16 + 23219*x^8 + 1)
 
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 + 257*x^32 + 14016*x^24 + 105419*x^16 + 23219*x^8 + 1, 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 + 257*x^32 + 14016*x^24 + 105419*x^16 + 23219*x^8 + 1);
 
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 + 257*x^32 + 14016*x^24 + 105419*x^16 + 23219*x^8 + 1);
 
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

$C_2\times C_{20}$ (as 40T2):

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 $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), 4.0.2048.2, \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{16})\), 10.0.219503494144.1, 10.10.7024111812608.1, 10.0.7024111812608.1, 20.0.50522262278163705147147943936.1, 20.0.1655513490330868290261743826894848.1, 20.20.1655513490330868290261743826894848.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $20^{2}$ $20^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{4}$ R $20^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ $20^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{10}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$

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 $40$$8$$5$$120$
\(11\) Copy content Toggle raw display 11.20.16.1$x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1644 x^{15} + 6440 x^{14} + 18720 x^{13} + 39010 x^{12} + 137400 x^{11} + 283734 x^{10} + 677980 x^{9} + 845540 x^{8} + 1499320 x^{7} + 2045360 x^{6} + 2492700 x^{5} + 5064740 x^{4} + 1884960 x^{3} + 9207500 x^{2} - 2109440 x + 7196441$$5$$4$$16$20T1$[\ ]_{5}^{4}$
11.20.16.1$x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1644 x^{15} + 6440 x^{14} + 18720 x^{13} + 39010 x^{12} + 137400 x^{11} + 283734 x^{10} + 677980 x^{9} + 845540 x^{8} + 1499320 x^{7} + 2045360 x^{6} + 2492700 x^{5} + 5064740 x^{4} + 1884960 x^{3} + 9207500 x^{2} - 2109440 x + 7196441$$5$$4$$16$20T1$[\ ]_{5}^{4}$