Properties

Label 32.0.203...000.6
Degree $32$
Signature $[0, 16]$
Discriminant $2.033\times 10^{53}$
Root discriminant \(46.33\)
Ramified primes $2,3,5$
Class number $2560$ (GRH)
Class group [4, 8, 80] (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 16*x^30 + 152*x^28 + 960*x^26 + 4525*x^24 + 16184*x^22 + 45376*x^20 + 98960*x^18 + 169224*x^16 + 220120*x^14 + 215964*x^12 + 146912*x^10 + 67325*x^8 + 13840*x^6 + 1988*x^4 + 48*x^2 + 1)
 
gp: K = bnfinit(y^32 + 16*y^30 + 152*y^28 + 960*y^26 + 4525*y^24 + 16184*y^22 + 45376*y^20 + 98960*y^18 + 169224*y^16 + 220120*y^14 + 215964*y^12 + 146912*y^10 + 67325*y^8 + 13840*y^6 + 1988*y^4 + 48*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 16*x^30 + 152*x^28 + 960*x^26 + 4525*x^24 + 16184*x^22 + 45376*x^20 + 98960*x^18 + 169224*x^16 + 220120*x^14 + 215964*x^12 + 146912*x^10 + 67325*x^8 + 13840*x^6 + 1988*x^4 + 48*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 16*x^30 + 152*x^28 + 960*x^26 + 4525*x^24 + 16184*x^22 + 45376*x^20 + 98960*x^18 + 169224*x^16 + 220120*x^14 + 215964*x^12 + 146912*x^10 + 67325*x^8 + 13840*x^6 + 1988*x^4 + 48*x^2 + 1)
 

\( x^{32} + 16 x^{30} + 152 x^{28} + 960 x^{26} + 4525 x^{24} + 16184 x^{22} + 45376 x^{20} + 98960 x^{18} + 169224 x^{16} + 220120 x^{14} + 215964 x^{12} + 146912 x^{10} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(203282392447840896882957090816000000000000000000000000\) \(\medspace = 2^{96}\cdot 3^{16}\cdot 5^{24}\) 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:  \(46.33\)
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:  $2^{3}3^{1/2}5^{3/4}\approx 46.331687411530766$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(240=2^{4}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(7,·)$, $\chi_{240}(143,·)$, $\chi_{240}(149,·)$, $\chi_{240}(23,·)$, $\chi_{240}(29,·)$, $\chi_{240}(161,·)$, $\chi_{240}(163,·)$, $\chi_{240}(167,·)$, $\chi_{240}(41,·)$, $\chi_{240}(43,·)$, $\chi_{240}(47,·)$, $\chi_{240}(49,·)$, $\chi_{240}(181,·)$, $\chi_{240}(187,·)$, $\chi_{240}(61,·)$, $\chi_{240}(67,·)$, $\chi_{240}(203,·)$, $\chi_{240}(209,·)$, $\chi_{240}(83,·)$, $\chi_{240}(89,·)$, $\chi_{240}(221,·)$, $\chi_{240}(223,·)$, $\chi_{240}(227,·)$, $\chi_{240}(101,·)$, $\chi_{240}(103,·)$, $\chi_{240}(107,·)$, $\chi_{240}(109,·)$, $\chi_{240}(229,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$, $\chi_{240}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4}a^{24}+\frac{1}{4}$, $\frac{1}{4}a^{25}+\frac{1}{4}a$, $\frac{1}{4}a^{26}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{27}+\frac{1}{4}a^{3}$, $\frac{1}{796}a^{28}+\frac{21}{398}a^{26}-\frac{37}{398}a^{24}+\frac{49}{199}a^{22}-\frac{27}{199}a^{20}+\frac{54}{199}a^{18}-\frac{23}{199}a^{16}-\frac{66}{199}a^{14}+\frac{60}{199}a^{12}+\frac{99}{199}a^{10}-\frac{28}{199}a^{8}+\frac{43}{199}a^{6}-\frac{283}{796}a^{4}+\frac{139}{398}a^{2}+\frac{65}{398}$, $\frac{1}{796}a^{29}+\frac{21}{398}a^{27}-\frac{37}{398}a^{25}+\frac{49}{199}a^{23}-\frac{27}{199}a^{21}+\frac{54}{199}a^{19}-\frac{23}{199}a^{17}-\frac{66}{199}a^{15}+\frac{60}{199}a^{13}+\frac{99}{199}a^{11}-\frac{28}{199}a^{9}+\frac{43}{199}a^{7}-\frac{283}{796}a^{5}+\frac{139}{398}a^{3}+\frac{65}{398}a$, $\frac{1}{33\!\cdots\!36}a^{30}+\frac{87\!\cdots\!73}{33\!\cdots\!36}a^{28}+\frac{19\!\cdots\!25}{16\!\cdots\!18}a^{26}+\frac{29\!\cdots\!75}{33\!\cdots\!36}a^{24}-\frac{34\!\cdots\!30}{83\!\cdots\!59}a^{22}-\frac{13\!\cdots\!91}{83\!\cdots\!59}a^{20}-\frac{21\!\cdots\!01}{83\!\cdots\!59}a^{18}+\frac{24\!\cdots\!97}{83\!\cdots\!59}a^{16}+\frac{24\!\cdots\!50}{83\!\cdots\!59}a^{14}+\frac{28\!\cdots\!21}{83\!\cdots\!59}a^{12}-\frac{38\!\cdots\!57}{83\!\cdots\!59}a^{10}+\frac{64\!\cdots\!53}{83\!\cdots\!59}a^{8}+\frac{15\!\cdots\!09}{33\!\cdots\!36}a^{6}+\frac{16\!\cdots\!41}{33\!\cdots\!36}a^{4}-\frac{72\!\cdots\!85}{16\!\cdots\!18}a^{2}+\frac{16\!\cdots\!91}{33\!\cdots\!36}$, $\frac{1}{33\!\cdots\!36}a^{31}+\frac{87\!\cdots\!73}{33\!\cdots\!36}a^{29}+\frac{19\!\cdots\!25}{16\!\cdots\!18}a^{27}+\frac{29\!\cdots\!75}{33\!\cdots\!36}a^{25}-\frac{34\!\cdots\!30}{83\!\cdots\!59}a^{23}-\frac{13\!\cdots\!91}{83\!\cdots\!59}a^{21}-\frac{21\!\cdots\!01}{83\!\cdots\!59}a^{19}+\frac{24\!\cdots\!97}{83\!\cdots\!59}a^{17}+\frac{24\!\cdots\!50}{83\!\cdots\!59}a^{15}+\frac{28\!\cdots\!21}{83\!\cdots\!59}a^{13}-\frac{38\!\cdots\!57}{83\!\cdots\!59}a^{11}+\frac{64\!\cdots\!53}{83\!\cdots\!59}a^{9}+\frac{15\!\cdots\!09}{33\!\cdots\!36}a^{7}+\frac{16\!\cdots\!41}{33\!\cdots\!36}a^{5}-\frac{72\!\cdots\!85}{16\!\cdots\!18}a^{3}+\frac{16\!\cdots\!91}{33\!\cdots\!36}a$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{4}\times C_{8}\times C_{80}$, which has order $2560$ (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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{4366251400675137246941}{166798748783521780363118} a^{30} - \frac{139419596465868338323055}{333597497567043560726236} a^{28} - \frac{330649311330808604060600}{83399374391760890181559} a^{26} - \frac{4169267722004324704631585}{166798748783521780363118} a^{24} - \frac{9808880326525928344379500}{83399374391760890181559} a^{22} - \frac{35006374090433504431991038}{83399374391760890181559} a^{20} - \frac{97913368924490331152866680}{83399374391760890181559} a^{18} - \frac{212870332447547083767334180}{83399374391760890181559} a^{16} - \frac{362653308169359271622909280}{83399374391760890181559} a^{14} - \frac{469216208282469529517110860}{83399374391760890181559} a^{12} - \frac{457227468859736880583674912}{83399374391760890181559} a^{10} - \frac{307351339084823514267820220}{83399374391760890181559} a^{8} - \frac{277212667649755983508367585}{166798748783521780363118} a^{6} - \frac{107263433800862459772757275}{333597497567043560726236} a^{4} - \frac{4042882751195651204983510}{83399374391760890181559} a^{2} - \frac{28386804048725376514543}{166798748783521780363118} \)  (order $6$) 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:   $\frac{10\!\cdots\!91}{83\!\cdots\!59}a^{30}+\frac{33\!\cdots\!61}{16\!\cdots\!18}a^{28}+\frac{15\!\cdots\!28}{83\!\cdots\!59}a^{26}+\frac{10\!\cdots\!53}{83\!\cdots\!59}a^{24}+\frac{47\!\cdots\!16}{83\!\cdots\!59}a^{22}+\frac{17\!\cdots\!44}{83\!\cdots\!59}a^{20}+\frac{48\!\cdots\!72}{83\!\cdots\!59}a^{18}+\frac{10\!\cdots\!17}{83\!\cdots\!59}a^{16}+\frac{18\!\cdots\!84}{83\!\cdots\!59}a^{14}+\frac{24\!\cdots\!96}{83\!\cdots\!59}a^{12}+\frac{24\!\cdots\!20}{83\!\cdots\!59}a^{10}+\frac{17\!\cdots\!68}{83\!\cdots\!59}a^{8}+\frac{84\!\cdots\!11}{83\!\cdots\!59}a^{6}+\frac{41\!\cdots\!97}{16\!\cdots\!18}a^{4}+\frac{25\!\cdots\!68}{83\!\cdots\!59}a^{2}+\frac{62\!\cdots\!85}{83\!\cdots\!59}$, $\frac{85\!\cdots\!79}{33\!\cdots\!36}a^{30}+\frac{34\!\cdots\!35}{83\!\cdots\!59}a^{28}+\frac{32\!\cdots\!87}{83\!\cdots\!59}a^{26}+\frac{20\!\cdots\!64}{83\!\cdots\!59}a^{24}+\frac{96\!\cdots\!13}{83\!\cdots\!59}a^{22}+\frac{34\!\cdots\!62}{83\!\cdots\!59}a^{20}+\frac{97\!\cdots\!20}{83\!\cdots\!59}a^{18}+\frac{21\!\cdots\!90}{83\!\cdots\!59}a^{16}+\frac{36\!\cdots\!78}{83\!\cdots\!59}a^{14}+\frac{47\!\cdots\!98}{83\!\cdots\!59}a^{12}+\frac{47\!\cdots\!87}{83\!\cdots\!59}a^{10}+\frac{32\!\cdots\!32}{83\!\cdots\!59}a^{8}+\frac{60\!\cdots\!67}{33\!\cdots\!36}a^{6}+\frac{32\!\cdots\!67}{83\!\cdots\!59}a^{4}+\frac{45\!\cdots\!16}{83\!\cdots\!59}a^{2}+\frac{10\!\cdots\!96}{83\!\cdots\!59}$, $\frac{17\!\cdots\!07}{33\!\cdots\!36}a^{30}+\frac{13\!\cdots\!23}{16\!\cdots\!18}a^{28}+\frac{13\!\cdots\!25}{16\!\cdots\!18}a^{26}+\frac{81\!\cdots\!45}{16\!\cdots\!18}a^{24}+\frac{19\!\cdots\!66}{83\!\cdots\!59}a^{22}+\frac{68\!\cdots\!48}{83\!\cdots\!59}a^{20}+\frac{19\!\cdots\!60}{83\!\cdots\!59}a^{18}+\frac{41\!\cdots\!48}{83\!\cdots\!59}a^{16}+\frac{71\!\cdots\!60}{83\!\cdots\!59}a^{14}+\frac{92\!\cdots\!70}{83\!\cdots\!59}a^{12}+\frac{90\!\cdots\!48}{83\!\cdots\!59}a^{10}+\frac{60\!\cdots\!20}{83\!\cdots\!59}a^{8}+\frac{10\!\cdots\!35}{33\!\cdots\!36}a^{6}+\frac{10\!\cdots\!87}{16\!\cdots\!18}a^{4}+\frac{15\!\cdots\!15}{16\!\cdots\!18}a^{2}+\frac{38\!\cdots\!81}{16\!\cdots\!18}$, $\frac{80\!\cdots\!25}{33\!\cdots\!36}a^{31}+\frac{64\!\cdots\!71}{16\!\cdots\!18}a^{29}+\frac{61\!\cdots\!89}{16\!\cdots\!18}a^{27}+\frac{38\!\cdots\!63}{16\!\cdots\!18}a^{25}+\frac{90\!\cdots\!90}{83\!\cdots\!59}a^{23}+\frac{32\!\cdots\!36}{83\!\cdots\!59}a^{21}+\frac{91\!\cdots\!60}{83\!\cdots\!59}a^{19}+\frac{19\!\cdots\!07}{83\!\cdots\!59}a^{17}+\frac{33\!\cdots\!48}{83\!\cdots\!59}a^{15}+\frac{44\!\cdots\!58}{83\!\cdots\!59}a^{13}+\frac{43\!\cdots\!59}{83\!\cdots\!59}a^{11}+\frac{29\!\cdots\!12}{83\!\cdots\!59}a^{9}+\frac{53\!\cdots\!97}{33\!\cdots\!36}a^{7}+\frac{54\!\cdots\!49}{16\!\cdots\!18}a^{5}+\frac{79\!\cdots\!87}{16\!\cdots\!18}a^{3}+\frac{19\!\cdots\!75}{16\!\cdots\!18}a$, $\frac{25\!\cdots\!89}{33\!\cdots\!36}a^{31}+\frac{41\!\cdots\!01}{33\!\cdots\!36}a^{29}+\frac{19\!\cdots\!25}{16\!\cdots\!18}a^{27}+\frac{61\!\cdots\!65}{83\!\cdots\!59}a^{25}+\frac{29\!\cdots\!66}{83\!\cdots\!59}a^{23}+\frac{10\!\cdots\!86}{83\!\cdots\!59}a^{21}+\frac{29\!\cdots\!40}{83\!\cdots\!59}a^{19}+\frac{63\!\cdots\!28}{83\!\cdots\!59}a^{17}+\frac{10\!\cdots\!40}{83\!\cdots\!59}a^{15}+\frac{13\!\cdots\!30}{83\!\cdots\!59}a^{13}+\frac{13\!\cdots\!60}{83\!\cdots\!59}a^{11}+\frac{91\!\cdots\!40}{83\!\cdots\!59}a^{9}+\frac{16\!\cdots\!05}{33\!\cdots\!36}a^{7}+\frac{32\!\cdots\!49}{33\!\cdots\!36}a^{5}+\frac{24\!\cdots\!35}{16\!\cdots\!18}a^{3}+\frac{29\!\cdots\!21}{83\!\cdots\!59}a$, $\frac{13\!\cdots\!05}{16\!\cdots\!18}a^{31}+\frac{10\!\cdots\!50}{83\!\cdots\!59}a^{29}+\frac{99\!\cdots\!84}{83\!\cdots\!59}a^{27}+\frac{12\!\cdots\!89}{16\!\cdots\!18}a^{25}+\frac{29\!\cdots\!48}{83\!\cdots\!59}a^{23}+\frac{10\!\cdots\!44}{83\!\cdots\!59}a^{21}+\frac{28\!\cdots\!38}{83\!\cdots\!59}a^{19}+\frac{61\!\cdots\!64}{83\!\cdots\!59}a^{17}+\frac{10\!\cdots\!88}{83\!\cdots\!59}a^{15}+\frac{12\!\cdots\!23}{83\!\cdots\!59}a^{13}+\frac{12\!\cdots\!60}{83\!\cdots\!59}a^{11}+\frac{76\!\cdots\!08}{83\!\cdots\!59}a^{9}+\frac{60\!\cdots\!49}{16\!\cdots\!18}a^{7}+\frac{27\!\cdots\!98}{83\!\cdots\!59}a^{5}+\frac{65\!\cdots\!48}{83\!\cdots\!59}a^{3}-\frac{20\!\cdots\!15}{16\!\cdots\!18}a$, $\frac{82\!\cdots\!69}{16\!\cdots\!18}a^{31}+\frac{26\!\cdots\!01}{33\!\cdots\!36}a^{29}+\frac{62\!\cdots\!65}{83\!\cdots\!59}a^{27}+\frac{38\!\cdots\!58}{83\!\cdots\!59}a^{25}+\frac{18\!\cdots\!36}{83\!\cdots\!59}a^{23}+\frac{64\!\cdots\!42}{83\!\cdots\!59}a^{21}+\frac{17\!\cdots\!79}{83\!\cdots\!59}a^{19}+\frac{38\!\cdots\!23}{83\!\cdots\!59}a^{17}+\frac{64\!\cdots\!80}{83\!\cdots\!59}a^{15}+\frac{80\!\cdots\!79}{83\!\cdots\!59}a^{13}+\frac{76\!\cdots\!84}{83\!\cdots\!59}a^{11}+\frac{47\!\cdots\!35}{83\!\cdots\!59}a^{9}+\frac{37\!\cdots\!49}{16\!\cdots\!18}a^{7}+\frac{67\!\cdots\!29}{33\!\cdots\!36}a^{5}+\frac{41\!\cdots\!99}{83\!\cdots\!59}a^{3}-\frac{89\!\cdots\!41}{83\!\cdots\!59}a$, $\frac{19\!\cdots\!61}{15\!\cdots\!58}a^{31}+\frac{62\!\cdots\!41}{30\!\cdots\!16}a^{29}+\frac{29\!\cdots\!65}{15\!\cdots\!58}a^{27}+\frac{37\!\cdots\!41}{30\!\cdots\!16}a^{25}+\frac{44\!\cdots\!86}{75\!\cdots\!79}a^{23}+\frac{15\!\cdots\!42}{75\!\cdots\!79}a^{21}+\frac{44\!\cdots\!92}{75\!\cdots\!79}a^{19}+\frac{96\!\cdots\!98}{75\!\cdots\!79}a^{17}+\frac{16\!\cdots\!40}{75\!\cdots\!79}a^{15}+\frac{21\!\cdots\!09}{75\!\cdots\!79}a^{13}+\frac{20\!\cdots\!04}{75\!\cdots\!79}a^{11}+\frac{13\!\cdots\!70}{75\!\cdots\!79}a^{9}+\frac{12\!\cdots\!13}{15\!\cdots\!58}a^{7}+\frac{48\!\cdots\!13}{30\!\cdots\!16}a^{5}+\frac{35\!\cdots\!23}{15\!\cdots\!58}a^{3}+\frac{25\!\cdots\!13}{30\!\cdots\!16}a+1$, $\frac{10\!\cdots\!65}{16\!\cdots\!18}a^{31}-\frac{43\!\cdots\!41}{16\!\cdots\!18}a^{30}+\frac{33\!\cdots\!65}{33\!\cdots\!36}a^{29}-\frac{13\!\cdots\!55}{33\!\cdots\!36}a^{28}+\frac{31\!\cdots\!25}{33\!\cdots\!36}a^{27}-\frac{33\!\cdots\!00}{83\!\cdots\!59}a^{26}+\frac{49\!\cdots\!81}{83\!\cdots\!59}a^{25}-\frac{41\!\cdots\!85}{16\!\cdots\!18}a^{24}+\frac{23\!\cdots\!65}{83\!\cdots\!59}a^{23}-\frac{98\!\cdots\!00}{83\!\cdots\!59}a^{22}+\frac{81\!\cdots\!30}{83\!\cdots\!59}a^{21}-\frac{35\!\cdots\!38}{83\!\cdots\!59}a^{20}+\frac{22\!\cdots\!80}{83\!\cdots\!59}a^{19}-\frac{97\!\cdots\!80}{83\!\cdots\!59}a^{18}+\frac{48\!\cdots\!45}{83\!\cdots\!59}a^{17}-\frac{21\!\cdots\!80}{83\!\cdots\!59}a^{16}+\frac{81\!\cdots\!00}{83\!\cdots\!59}a^{15}-\frac{36\!\cdots\!80}{83\!\cdots\!59}a^{14}+\frac{10\!\cdots\!85}{83\!\cdots\!59}a^{13}-\frac{46\!\cdots\!60}{83\!\cdots\!59}a^{12}+\frac{96\!\cdots\!10}{83\!\cdots\!59}a^{11}-\frac{45\!\cdots\!12}{83\!\cdots\!59}a^{10}+\frac{60\!\cdots\!25}{83\!\cdots\!59}a^{9}-\frac{30\!\cdots\!20}{83\!\cdots\!59}a^{8}+\frac{48\!\cdots\!95}{16\!\cdots\!18}a^{7}-\frac{27\!\cdots\!85}{16\!\cdots\!18}a^{6}+\frac{86\!\cdots\!85}{33\!\cdots\!36}a^{5}-\frac{10\!\cdots\!75}{33\!\cdots\!36}a^{4}+\frac{20\!\cdots\!65}{33\!\cdots\!36}a^{3}-\frac{40\!\cdots\!10}{83\!\cdots\!59}a^{2}-\frac{97\!\cdots\!70}{83\!\cdots\!59}a-\frac{19\!\cdots\!61}{16\!\cdots\!18}$, $\frac{26\!\cdots\!79}{75\!\cdots\!79}a^{31}-\frac{87589632705}{8243253718082}a^{30}+\frac{16\!\cdots\!03}{30\!\cdots\!16}a^{29}-\frac{13940291661}{82846771036}a^{28}+\frac{39\!\cdots\!35}{75\!\cdots\!79}a^{27}-\frac{6542359560696}{4121626859041}a^{26}+\frac{10\!\cdots\!73}{30\!\cdots\!16}a^{25}-\frac{81919443833915}{8243253718082}a^{24}+\frac{11\!\cdots\!98}{75\!\cdots\!79}a^{23}-\frac{191360253669396}{4121626859041}a^{22}+\frac{42\!\cdots\!11}{75\!\cdots\!79}a^{21}-\frac{676822404992814}{4121626859041}a^{20}+\frac{11\!\cdots\!26}{75\!\cdots\!79}a^{19}-\frac{18\!\cdots\!28}{4121626859041}a^{18}+\frac{25\!\cdots\!34}{75\!\cdots\!79}a^{17}-\frac{40\!\cdots\!53}{4121626859041}a^{16}+\frac{43\!\cdots\!91}{75\!\cdots\!79}a^{15}-\frac{67\!\cdots\!72}{4121626859041}a^{14}+\frac{56\!\cdots\!17}{75\!\cdots\!79}a^{13}-\frac{84\!\cdots\!20}{4121626859041}a^{12}+\frac{55\!\cdots\!42}{75\!\cdots\!79}a^{11}-\frac{79\!\cdots\!16}{4121626859041}a^{10}+\frac{37\!\cdots\!35}{75\!\cdots\!79}a^{9}-\frac{49\!\cdots\!97}{4121626859041}a^{8}+\frac{16\!\cdots\!07}{75\!\cdots\!79}a^{7}-\frac{39\!\cdots\!61}{8243253718082}a^{6}+\frac{13\!\cdots\!59}{30\!\cdots\!16}a^{5}-\frac{709863284825799}{16486507436164}a^{4}+\frac{44\!\cdots\!97}{75\!\cdots\!79}a^{3}-\frac{4298131170078}{4121626859041}a^{2}+\frac{69\!\cdots\!09}{30\!\cdots\!16}a+\frac{13181914833877}{8243253718082}$, $\frac{15\!\cdots\!23}{83\!\cdots\!59}a^{31}+\frac{96\!\cdots\!13}{33\!\cdots\!36}a^{29}+\frac{91\!\cdots\!01}{33\!\cdots\!36}a^{27}+\frac{57\!\cdots\!91}{33\!\cdots\!36}a^{25}+\frac{68\!\cdots\!42}{83\!\cdots\!59}a^{23}+\frac{24\!\cdots\!54}{83\!\cdots\!59}a^{21}+\frac{68\!\cdots\!62}{83\!\cdots\!59}a^{19}+\frac{14\!\cdots\!94}{83\!\cdots\!59}a^{17}+\frac{25\!\cdots\!69}{83\!\cdots\!59}a^{15}+\frac{32\!\cdots\!35}{83\!\cdots\!59}a^{13}+\frac{31\!\cdots\!66}{83\!\cdots\!59}a^{11}+\frac{21\!\cdots\!54}{83\!\cdots\!59}a^{9}+\frac{96\!\cdots\!79}{83\!\cdots\!59}a^{7}+\frac{74\!\cdots\!81}{33\!\cdots\!36}a^{5}+\frac{10\!\cdots\!45}{33\!\cdots\!36}a^{3}+\frac{39\!\cdots\!51}{33\!\cdots\!36}a+1$, $\frac{58\!\cdots\!83}{16\!\cdots\!18}a^{31}-\frac{43\!\cdots\!41}{16\!\cdots\!18}a^{30}+\frac{18\!\cdots\!69}{33\!\cdots\!36}a^{29}-\frac{13\!\cdots\!55}{33\!\cdots\!36}a^{28}+\frac{87\!\cdots\!53}{16\!\cdots\!18}a^{27}-\frac{33\!\cdots\!00}{83\!\cdots\!59}a^{26}+\frac{11\!\cdots\!57}{33\!\cdots\!36}a^{25}-\frac{41\!\cdots\!85}{16\!\cdots\!18}a^{24}+\frac{12\!\cdots\!22}{83\!\cdots\!59}a^{23}-\frac{98\!\cdots\!00}{83\!\cdots\!59}a^{22}+\frac{45\!\cdots\!62}{83\!\cdots\!59}a^{21}-\frac{35\!\cdots\!38}{83\!\cdots\!59}a^{20}+\frac{12\!\cdots\!91}{83\!\cdots\!59}a^{19}-\frac{97\!\cdots\!80}{83\!\cdots\!59}a^{18}+\frac{27\!\cdots\!71}{83\!\cdots\!59}a^{17}-\frac{21\!\cdots\!80}{83\!\cdots\!59}a^{16}+\frac{45\!\cdots\!48}{83\!\cdots\!59}a^{15}-\frac{36\!\cdots\!80}{83\!\cdots\!59}a^{14}+\frac{57\!\cdots\!33}{83\!\cdots\!59}a^{13}-\frac{46\!\cdots\!60}{83\!\cdots\!59}a^{12}+\frac{53\!\cdots\!56}{83\!\cdots\!59}a^{11}-\frac{45\!\cdots\!12}{83\!\cdots\!59}a^{10}+\frac{33\!\cdots\!63}{83\!\cdots\!59}a^{9}-\frac{30\!\cdots\!20}{83\!\cdots\!59}a^{8}+\frac{26\!\cdots\!63}{16\!\cdots\!18}a^{7}-\frac{27\!\cdots\!85}{16\!\cdots\!18}a^{6}+\frac{47\!\cdots\!53}{33\!\cdots\!36}a^{5}-\frac{10\!\cdots\!75}{33\!\cdots\!36}a^{4}+\frac{57\!\cdots\!13}{16\!\cdots\!18}a^{3}-\frac{40\!\cdots\!10}{83\!\cdots\!59}a^{2}-\frac{32\!\cdots\!19}{33\!\cdots\!36}a-\frac{19\!\cdots\!61}{16\!\cdots\!18}$, $\frac{22\!\cdots\!11}{10\!\cdots\!21}a^{31}-\frac{43\!\cdots\!41}{16\!\cdots\!18}a^{30}+\frac{17\!\cdots\!13}{53\!\cdots\!79}a^{29}-\frac{13\!\cdots\!55}{33\!\cdots\!36}a^{28}+\frac{13\!\cdots\!57}{42\!\cdots\!84}a^{27}-\frac{33\!\cdots\!00}{83\!\cdots\!59}a^{26}+\frac{21\!\cdots\!56}{10\!\cdots\!21}a^{25}-\frac{41\!\cdots\!85}{16\!\cdots\!18}a^{24}+\frac{99\!\cdots\!60}{10\!\cdots\!21}a^{23}-\frac{98\!\cdots\!00}{83\!\cdots\!59}a^{22}+\frac{35\!\cdots\!00}{10\!\cdots\!21}a^{21}-\frac{35\!\cdots\!38}{83\!\cdots\!59}a^{20}+\frac{99\!\cdots\!10}{10\!\cdots\!21}a^{19}-\frac{97\!\cdots\!80}{83\!\cdots\!59}a^{18}+\frac{21\!\cdots\!23}{10\!\cdots\!21}a^{17}-\frac{21\!\cdots\!80}{83\!\cdots\!59}a^{16}+\frac{37\!\cdots\!07}{10\!\cdots\!21}a^{15}-\frac{36\!\cdots\!80}{83\!\cdots\!59}a^{14}+\frac{48\!\cdots\!92}{10\!\cdots\!21}a^{13}-\frac{46\!\cdots\!60}{83\!\cdots\!59}a^{12}+\frac{47\!\cdots\!54}{10\!\cdots\!21}a^{11}-\frac{45\!\cdots\!12}{83\!\cdots\!59}a^{10}+\frac{31\!\cdots\!68}{10\!\cdots\!21}a^{9}-\frac{30\!\cdots\!20}{83\!\cdots\!59}a^{8}+\frac{14\!\cdots\!22}{10\!\cdots\!21}a^{7}-\frac{27\!\cdots\!85}{16\!\cdots\!18}a^{6}+\frac{29\!\cdots\!28}{10\!\cdots\!21}a^{5}-\frac{10\!\cdots\!75}{33\!\cdots\!36}a^{4}+\frac{17\!\cdots\!41}{42\!\cdots\!84}a^{3}-\frac{40\!\cdots\!10}{83\!\cdots\!59}a^{2}+\frac{10\!\cdots\!26}{10\!\cdots\!21}a-\frac{28\!\cdots\!43}{16\!\cdots\!18}$, $\frac{77\!\cdots\!08}{83\!\cdots\!59}a^{31}-\frac{70\!\cdots\!95}{33\!\cdots\!36}a^{30}+\frac{49\!\cdots\!89}{33\!\cdots\!36}a^{29}-\frac{11\!\cdots\!95}{33\!\cdots\!36}a^{28}+\frac{47\!\cdots\!59}{33\!\cdots\!36}a^{27}-\frac{26\!\cdots\!69}{83\!\cdots\!59}a^{26}+\frac{29\!\cdots\!79}{33\!\cdots\!36}a^{25}-\frac{65\!\cdots\!97}{33\!\cdots\!36}a^{24}+\frac{35\!\cdots\!14}{83\!\cdots\!59}a^{23}-\frac{76\!\cdots\!88}{83\!\cdots\!59}a^{22}+\frac{12\!\cdots\!78}{83\!\cdots\!59}a^{21}-\frac{26\!\cdots\!94}{83\!\cdots\!59}a^{20}+\frac{35\!\cdots\!78}{83\!\cdots\!59}a^{19}-\frac{74\!\cdots\!96}{83\!\cdots\!59}a^{18}+\frac{76\!\cdots\!22}{83\!\cdots\!59}a^{17}-\frac{15\!\cdots\!09}{83\!\cdots\!59}a^{16}+\frac{13\!\cdots\!92}{83\!\cdots\!59}a^{15}-\frac{26\!\cdots\!08}{83\!\cdots\!59}a^{14}+\frac{16\!\cdots\!51}{83\!\cdots\!59}a^{13}-\frac{33\!\cdots\!53}{83\!\cdots\!59}a^{12}+\frac{16\!\cdots\!10}{83\!\cdots\!59}a^{11}-\frac{31\!\cdots\!40}{83\!\cdots\!59}a^{10}+\frac{11\!\cdots\!54}{83\!\cdots\!59}a^{9}-\frac{19\!\cdots\!53}{83\!\cdots\!59}a^{8}+\frac{50\!\cdots\!36}{83\!\cdots\!59}a^{7}-\frac{31\!\cdots\!51}{33\!\cdots\!36}a^{6}+\frac{38\!\cdots\!37}{33\!\cdots\!36}a^{5}-\frac{28\!\cdots\!27}{33\!\cdots\!36}a^{4}+\frac{52\!\cdots\!71}{33\!\cdots\!36}a^{3}-\frac{17\!\cdots\!23}{83\!\cdots\!59}a^{2}+\frac{20\!\cdots\!99}{33\!\cdots\!36}a+\frac{17\!\cdots\!23}{33\!\cdots\!36}$, $\frac{43\!\cdots\!41}{16\!\cdots\!18}a^{31}+\frac{87589632705}{8243253718082}a^{30}+\frac{13\!\cdots\!55}{33\!\cdots\!36}a^{29}+\frac{13940291661}{82846771036}a^{28}+\frac{33\!\cdots\!00}{83\!\cdots\!59}a^{27}+\frac{6542359560696}{4121626859041}a^{26}+\frac{41\!\cdots\!85}{16\!\cdots\!18}a^{25}+\frac{81919443833915}{8243253718082}a^{24}+\frac{98\!\cdots\!00}{83\!\cdots\!59}a^{23}+\frac{191360253669396}{4121626859041}a^{22}+\frac{35\!\cdots\!38}{83\!\cdots\!59}a^{21}+\frac{676822404992814}{4121626859041}a^{20}+\frac{97\!\cdots\!80}{83\!\cdots\!59}a^{19}+\frac{18\!\cdots\!28}{4121626859041}a^{18}+\frac{21\!\cdots\!80}{83\!\cdots\!59}a^{17}+\frac{40\!\cdots\!53}{4121626859041}a^{16}+\frac{36\!\cdots\!80}{83\!\cdots\!59}a^{15}+\frac{67\!\cdots\!72}{4121626859041}a^{14}+\frac{46\!\cdots\!60}{83\!\cdots\!59}a^{13}+\frac{84\!\cdots\!20}{4121626859041}a^{12}+\frac{45\!\cdots\!12}{83\!\cdots\!59}a^{11}+\frac{79\!\cdots\!16}{4121626859041}a^{10}+\frac{30\!\cdots\!20}{83\!\cdots\!59}a^{9}+\frac{49\!\cdots\!97}{4121626859041}a^{8}+\frac{27\!\cdots\!85}{16\!\cdots\!18}a^{7}+\frac{39\!\cdots\!61}{8243253718082}a^{6}+\frac{10\!\cdots\!75}{33\!\cdots\!36}a^{5}+\frac{709863284825799}{16486507436164}a^{4}+\frac{40\!\cdots\!10}{83\!\cdots\!59}a^{3}+\frac{4298131170078}{4121626859041}a^{2}+\frac{28\!\cdots\!43}{16\!\cdots\!18}a-\frac{13181914833877}{8243253718082}$ 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:  \( 48613521256.81357 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 48613521256.81357 \cdot 2560}{6\cdot\sqrt{203282392447840896882957090816000000000000000000000000}}\cr\approx \mathstrut & 0.271439975078757 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 16*x^30 + 152*x^28 + 960*x^26 + 4525*x^24 + 16184*x^22 + 45376*x^20 + 98960*x^18 + 169224*x^16 + 220120*x^14 + 215964*x^12 + 146912*x^10 + 67325*x^8 + 13840*x^6 + 1988*x^4 + 48*x^2 + 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^32 + 16*x^30 + 152*x^28 + 960*x^26 + 4525*x^24 + 16184*x^22 + 45376*x^20 + 98960*x^18 + 169224*x^16 + 220120*x^14 + 215964*x^12 + 146912*x^10 + 67325*x^8 + 13840*x^6 + 1988*x^4 + 48*x^2 + 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^32 + 16*x^30 + 152*x^28 + 960*x^26 + 4525*x^24 + 16184*x^22 + 45376*x^20 + 98960*x^18 + 169224*x^16 + 220120*x^14 + 215964*x^12 + 146912*x^10 + 67325*x^8 + 13840*x^6 + 1988*x^4 + 48*x^2 + 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^32 + 16*x^30 + 152*x^28 + 960*x^26 + 4525*x^24 + 16184*x^22 + 45376*x^20 + 98960*x^18 + 169224*x^16 + 220120*x^14 + 215964*x^12 + 146912*x^10 + 67325*x^8 + 13840*x^6 + 1988*x^4 + 48*x^2 + 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

$C_2\times C_4^2$ (as 32T36):

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 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 4.0.18432.2, \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), 4.0.460800.2, 4.4.51200.1, \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.2304000.1, 4.4.256000.2, 4.0.2304000.2, 4.4.256000.1, \(\Q(\zeta_{20})^+\), 4.0.72000.2, 4.4.8000.1, 4.0.18000.1, 8.0.339738624.2, 8.0.207360000.1, 8.0.212336640000.7, 8.0.212336640000.5, 8.0.212336640000.2, 8.8.2621440000.1, 8.0.212336640000.4, 8.0.5308416000000.3, 8.0.5308416000000.9, 8.0.82944000000.2, 8.0.82944000000.3, 8.0.5308416000000.7, 8.8.65536000000.1, \(\Q(\zeta_{40})^+\), 8.0.82944000000.6, 8.0.5308416000000.2, 8.0.5308416000000.6, 8.0.324000000.2, 8.0.5184000000.4, 16.0.45086848686489600000000.1, 16.0.28179280429056000000000000.1, 16.0.6879707136000000000000.5, 16.0.450868486864896000000000000.7, 16.0.450868486864896000000000000.6, 16.0.450868486864896000000000000.13, \(\Q(\zeta_{80})^+\)

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 R R ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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 $16$$8$$2$$48$
Deg $16$$8$$2$$48$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$