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Properties

Label 32.0.550...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $5.500\times 10^{47}$
Root discriminant \(31.04\)
Ramified primes $2,5,29$
Class number $18$ (GRH)
Class group [3, 6] (GRH)
Galois group $C_2^4:C_4$ (as 32T262)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 6*x^28 + 315*x^24 - 2036*x^20 + 5109*x^16 - 2036*x^12 + 315*x^8 + 6*x^4 + 1)
 
gp: K = bnfinit(y^32 + 6*y^28 + 315*y^24 - 2036*y^20 + 5109*y^16 - 2036*y^12 + 315*y^8 + 6*y^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 6*x^28 + 315*x^24 - 2036*x^20 + 5109*x^16 - 2036*x^12 + 315*x^8 + 6*x^4 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 6*x^28 + 315*x^24 - 2036*x^20 + 5109*x^16 - 2036*x^12 + 315*x^8 + 6*x^4 + 1)
 

\( x^{32} + 6x^{28} + 315x^{24} - 2036x^{20} + 5109x^{16} - 2036x^{12} + 315x^{8} + 6x^{4} + 1 \) 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:   \(550026747803854214004736000000000000000000000000\) \(\medspace = 2^{64}\cdot 5^{24}\cdot 29^{8}\) 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:  \(31.04\)
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:  $2^{2}5^{3/4}29^{1/2}\approx 72.02553510942846$
Ramified primes:   \(2\), \(5\), \(29\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{8}a^{16}-\frac{1}{8}a^{12}+\frac{1}{8}a^{8}-\frac{1}{8}a^{4}+\frac{1}{8}$, $\frac{1}{8}a^{17}-\frac{1}{8}a^{13}+\frac{1}{8}a^{9}-\frac{1}{8}a^{5}+\frac{1}{8}a$, $\frac{1}{8}a^{18}-\frac{1}{8}a^{14}+\frac{1}{8}a^{10}-\frac{1}{8}a^{6}+\frac{1}{8}a^{2}$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{15}+\frac{1}{8}a^{11}-\frac{1}{8}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{264}a^{20}+\frac{1}{22}a^{16}-\frac{3}{22}a^{12}-\frac{3}{22}a^{8}+\frac{1}{22}a^{4}-\frac{131}{264}$, $\frac{1}{264}a^{21}+\frac{1}{22}a^{17}-\frac{3}{22}a^{13}-\frac{3}{22}a^{9}+\frac{1}{22}a^{5}-\frac{131}{264}a$, $\frac{1}{264}a^{22}+\frac{1}{22}a^{18}-\frac{3}{22}a^{14}-\frac{3}{22}a^{10}+\frac{1}{22}a^{6}-\frac{131}{264}a^{2}$, $\frac{1}{264}a^{23}+\frac{1}{22}a^{19}-\frac{3}{22}a^{15}-\frac{3}{22}a^{11}+\frac{1}{22}a^{7}-\frac{131}{264}a^{3}$, $\frac{1}{46728}a^{24}-\frac{7}{23364}a^{20}-\frac{36}{649}a^{16}+\frac{527}{1947}a^{12}-\frac{95}{649}a^{8}-\frac{5855}{46728}a^{4}+\frac{9293}{23364}$, $\frac{1}{46728}a^{25}-\frac{7}{23364}a^{21}-\frac{36}{649}a^{17}+\frac{527}{1947}a^{13}-\frac{95}{649}a^{9}-\frac{5855}{46728}a^{5}+\frac{9293}{23364}a$, $\frac{1}{46728}a^{26}-\frac{7}{23364}a^{22}-\frac{36}{649}a^{18}+\frac{527}{1947}a^{14}-\frac{95}{649}a^{10}-\frac{5855}{46728}a^{6}+\frac{9293}{23364}a^{2}$, $\frac{1}{46728}a^{27}-\frac{7}{23364}a^{23}-\frac{36}{649}a^{19}+\frac{527}{1947}a^{15}-\frac{95}{649}a^{11}-\frac{5855}{46728}a^{7}+\frac{9293}{23364}a^{3}$, $\frac{1}{5747544}a^{28}-\frac{17}{2873772}a^{24}-\frac{4967}{5747544}a^{20}+\frac{12068}{239481}a^{16}+\frac{25283}{239481}a^{12}+\frac{203161}{5747544}a^{8}-\frac{1104605}{2873772}a^{4}+\frac{92905}{5747544}$, $\frac{1}{5747544}a^{29}-\frac{17}{2873772}a^{25}-\frac{4967}{5747544}a^{21}+\frac{12068}{239481}a^{17}+\frac{25283}{239481}a^{13}+\frac{203161}{5747544}a^{9}-\frac{1104605}{2873772}a^{5}+\frac{92905}{5747544}a$, $\frac{1}{5747544}a^{30}-\frac{17}{2873772}a^{26}-\frac{4967}{5747544}a^{22}+\frac{12068}{239481}a^{18}+\frac{25283}{239481}a^{14}+\frac{203161}{5747544}a^{10}-\frac{1104605}{2873772}a^{6}+\frac{92905}{5747544}a^{2}$, $\frac{1}{5747544}a^{31}-\frac{17}{2873772}a^{27}-\frac{4967}{5747544}a^{23}+\frac{12068}{239481}a^{19}+\frac{25283}{239481}a^{15}+\frac{203161}{5747544}a^{11}-\frac{1104605}{2873772}a^{7}+\frac{92905}{5747544}a^{3}$ 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

$C_{3}\times C_{6}$, which has order $18$ (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:   \( -\frac{471593}{5747544} a^{29} - \frac{2788897}{5747544} a^{25} - \frac{74151091}{2873772} a^{21} + \frac{324334393}{1915848} a^{17} - \frac{830131385}{1915848} a^{13} + \frac{577825349}{2873772} a^{9} - \frac{844661}{24354} a^{5} - \frac{4121555}{5747544} a \)  (order $40$) 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:   $\frac{209389}{5747544}a^{28}+\frac{1242437}{5747544}a^{24}+\frac{65865709}{5747544}a^{20}-\frac{17947804}{239481}a^{16}+\frac{45640721}{239481}a^{12}-\frac{481175291}{5747544}a^{8}+\frac{54919445}{5747544}a^{4}-\frac{1387811}{5747544}$, $\frac{140947}{5747544}a^{28}+\frac{836651}{5747544}a^{24}+\frac{11084797}{1436886}a^{20}-\frac{8783173}{174168}a^{16}+\frac{245641375}{1915848}a^{12}-\frac{161853601}{2873772}a^{8}+\frac{5469712}{718443}a^{4}-\frac{933641}{5747544}$, $\frac{475001}{2873772}a^{30}+\frac{514261}{522504}a^{26}+\frac{27181883}{522504}a^{22}-\frac{649245233}{1915848}a^{18}+\frac{1648078705}{1915848}a^{14}-\frac{2171433977}{5747544}a^{10}+\frac{212611825}{2873772}a^{6}-\frac{3131473}{2873772}a^{2}-1$, $\frac{203897}{5747544}a^{30}+\frac{1254505}{5747544}a^{26}+\frac{64413641}{5747544}a^{22}-\frac{33777697}{478962}a^{18}+\frac{81512603}{478962}a^{14}-\frac{254131219}{5747544}a^{10}-\frac{6464867}{5747544}a^{6}+\frac{2224231}{522504}a^{2}-1$, $\frac{933641}{5747544}a^{31}+\frac{5742793}{5747544}a^{27}+\frac{2499437}{48708}a^{23}-\frac{77356412}{239481}a^{19}+\frac{186671965}{239481}a^{15}-\frac{1163968951}{5747544}a^{11}-\frac{29610287}{5747544}a^{7}+\frac{24679771}{2873772}a^{3}+1$, $\frac{45929}{957924}a^{29}+\frac{94157}{319308}a^{25}+\frac{29016863}{1915848}a^{21}-\frac{60895325}{638616}a^{17}+\frac{146947109}{638616}a^{13}-\frac{114533597}{1915848}a^{9}-\frac{971209}{638616}a^{5}+\frac{458593}{239481}a+1$, $\frac{77683}{212872}a^{31}+\frac{4225729}{1915848}a^{27}+\frac{220420543}{1915848}a^{23}-\frac{157079993}{212872}a^{19}+\frac{1170151891}{638616}a^{15}-\frac{70768127}{106436}a^{11}+\frac{87564365}{957924}a^{7}+\frac{6949103}{957924}a^{3}-1$, $\frac{298489}{5747544}a^{30}-\frac{45929}{957924}a^{29}+\frac{152563}{522504}a^{26}-\frac{94157}{319308}a^{25}+\frac{93335503}{5747544}a^{22}-\frac{29016863}{1915848}a^{21}-\frac{214437479}{1915848}a^{18}+\frac{60895325}{638616}a^{17}+\frac{583564087}{1915848}a^{14}-\frac{146947109}{638616}a^{13}-\frac{290014655}{1436886}a^{10}+\frac{114533597}{1915848}a^{9}+\frac{133193983}{2873772}a^{6}+\frac{971209}{638616}a^{5}-\frac{9432895}{2873772}a^{2}-\frac{458593}{239481}a$, $\frac{61667}{2873772}a^{28}+\frac{711515}{5747544}a^{24}+\frac{38677027}{5747544}a^{20}-\frac{10837528}{239481}a^{16}+\frac{28640015}{239481}a^{12}-\frac{196923433}{2873772}a^{8}+\frac{89693075}{5747544}a^{4}-\frac{6365789}{5747544}$, $\frac{8857}{53218}a^{30}+\frac{209389}{5747544}a^{29}+\frac{2410}{2419}a^{26}+\frac{1242437}{5747544}a^{25}+\frac{1394530}{26609}a^{22}+\frac{65865709}{5747544}a^{21}-\frac{9036150}{26609}a^{18}-\frac{17947804}{239481}a^{17}+\frac{22723590}{26609}a^{14}+\frac{45640721}{239481}a^{13}-\frac{18337285}{53218}a^{10}-\frac{481175291}{5747544}a^{9}+\frac{1136660}{26609}a^{6}+\frac{54919445}{5747544}a^{5}+\frac{90220}{26609}a^{2}-\frac{1387811}{5747544}a-1$, $\frac{75695}{957924}a^{30}+\frac{83731}{957924}a^{29}+\frac{54047}{1915848}a^{28}+\frac{465577}{957924}a^{26}+\frac{488783}{957924}a^{25}+\frac{335443}{1915848}a^{24}+\frac{5977889}{239481}a^{22}+\frac{6572881}{239481}a^{21}+\frac{17095757}{1915848}a^{20}-\frac{100351511}{638616}a^{18}-\frac{116516089}{638616}a^{17}-\frac{35499457}{638616}a^{16}+\frac{242161327}{638616}a^{14}+\frac{303204833}{638616}a^{13}+\frac{84890681}{638616}a^{12}-\frac{188745647}{1915848}a^{10}-\frac{471396613}{1915848}a^{9}-\frac{15141553}{478962}a^{8}-\frac{4801513}{1915848}a^{6}+\frac{86641033}{1915848}a^{5}+\frac{6300623}{957924}a^{4}+\frac{4628923}{1915848}a^{2}+\frac{1866917}{1915848}a-\frac{458159}{957924}$, $\frac{264239}{638616}a^{31}-\frac{285475}{5747544}a^{29}+\frac{471593}{5747544}a^{28}+\frac{590497}{239481}a^{27}-\frac{7279}{24354}a^{25}+\frac{2788897}{5747544}a^{24}+\frac{249507125}{1915848}a^{23}-\frac{44982149}{2873772}a^{21}+\frac{74151091}{2873772}a^{20}-\frac{90232357}{106436}a^{19}+\frac{193198595}{1915848}a^{17}-\frac{324334393}{1915848}a^{16}+\frac{685718095}{319308}a^{15}-\frac{483792163}{1915848}a^{13}+\frac{830131385}{1915848}a^{12}-\frac{589723495}{638616}a^{11}+\frac{287772751}{2873772}a^{9}-\frac{577825349}{2873772}a^{8}+\frac{145013701}{957924}a^{7}-\frac{127496813}{5747544}a^{5}+\frac{844661}{24354}a^{4}+\frac{5782043}{1915848}a^{3}+\frac{828613}{522504}a+\frac{4121555}{5747544}$, $\frac{2605}{14514}a^{31}+\frac{40781}{478962}a^{29}-\frac{285475}{5747544}a^{28}+\frac{1020035}{957924}a^{27}+\frac{110087}{212872}a^{25}-\frac{7279}{24354}a^{24}+\frac{4917445}{87084}a^{23}+\frac{51459313}{1915848}a^{21}-\frac{44982149}{2873772}a^{20}-\frac{9824070}{26609}a^{19}-\frac{109436461}{638616}a^{17}+\frac{193198595}{1915848}a^{16}+\frac{6840229}{7257}a^{15}+\frac{269947525}{638616}a^{13}-\frac{483792163}{1915848}a^{12}-\frac{69245785}{159654}a^{11}-\frac{276558907}{1915848}a^{9}+\frac{287772751}{2873772}a^{8}+\frac{8472505}{87084}a^{7}+\frac{6723997}{319308}a^{5}-\frac{127496813}{5747544}a^{4}-\frac{6640705}{957924}a^{3}+\frac{145519}{87084}a+\frac{828613}{522504}$, $\frac{2245931}{5747544}a^{31}-\frac{973979}{5747544}a^{30}+\frac{40781}{478962}a^{29}+\frac{1669718}{718443}a^{27}-\frac{520145}{522504}a^{26}+\frac{110087}{212872}a^{25}+\frac{353372341}{2873772}a^{23}-\frac{153025663}{2873772}a^{22}+\frac{51459313}{1915848}a^{21}-\frac{1536646825}{1915848}a^{19}+\frac{259585}{738}a^{18}-\frac{109436461}{638616}a^{17}+\frac{354822643}{174168}a^{15}-\frac{434936471}{478962}a^{14}+\frac{269947525}{638616}a^{13}-\frac{642856226}{718443}a^{11}+\frac{233621867}{522504}a^{10}-\frac{276558907}{1915848}a^{9}+\frac{874271059}{5747544}a^{7}-\frac{459263095}{5747544}a^{6}+\frac{6723997}{319308}a^{5}-\frac{14833177}{5747544}a^{3}-\frac{441923}{261252}a^{2}+\frac{145519}{87084}a$, $\frac{1063435}{5747544}a^{30}-\frac{471593}{5747544}a^{29}+\frac{27539}{718443}a^{28}+\frac{793171}{718443}a^{26}-\frac{2788897}{5747544}a^{25}+\frac{1307705}{5747544}a^{24}+\frac{334764031}{5747544}a^{22}-\frac{74151091}{2873772}a^{21}+\frac{69308869}{5747544}a^{20}-\frac{725432837}{1915848}a^{18}+\frac{324334393}{1915848}a^{17}-\frac{151017641}{1915848}a^{16}+\frac{1834247989}{1915848}a^{14}-\frac{830131385}{1915848}a^{13}+\frac{384318793}{1915848}a^{12}-\frac{583334033}{1436886}a^{10}+\frac{577825349}{2873772}a^{9}-\frac{514811507}{5747544}a^{8}+\frac{378679031}{5747544}a^{6}-\frac{844661}{24354}a^{5}+\frac{21653963}{1436886}a^{4}+\frac{936530}{718443}a^{2}-\frac{4121555}{5747544}a+\frac{879953}{2873772}$ 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:  \( 76396638760.7958 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 76396638760.7958 \cdot 18}{40\cdot\sqrt{550026747803854214004736000000000000000000000000}}\cr\approx \mathstrut & 0.273509713046900 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 6*x^28 + 315*x^24 - 2036*x^20 + 5109*x^16 - 2036*x^12 + 315*x^8 + 6*x^4 + 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 + 6*x^28 + 315*x^24 - 2036*x^20 + 5109*x^16 - 2036*x^12 + 315*x^8 + 6*x^4 + 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 + 6*x^28 + 315*x^24 - 2036*x^20 + 5109*x^16 - 2036*x^12 + 315*x^8 + 6*x^4 + 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 + 6*x^28 + 315*x^24 - 2036*x^20 + 5109*x^16 - 2036*x^12 + 315*x^8 + 6*x^4 + 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

$C_2^4:C_4$ (as 32T262):

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 64
The 40 conjugacy class representatives for $C_2^4:C_4$
Character table for $C_2^4:C_4$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), 4.0.232000.1, \(\Q(\zeta_{5})\), 4.4.58000.1, 4.4.8000.1, 4.0.3625.1, 4.0.8000.2, 4.4.232000.1, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{8})\), 4.4.46400.1, \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.46400.1, \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), 4.4.725.1, 4.0.11600.1, 8.0.34447360000.23, 8.8.2152960000.1, 8.0.40960000.1, 8.0.34447360000.3, 8.0.2152960000.5, 8.0.34447360000.26, 8.0.134560000.4, 8.0.53824000000.12, 8.8.861184000000.5, 8.0.53824000000.1, 8.8.861184000000.2, 8.0.861184000000.11, 8.0.64000000.1, 8.0.53824000000.14, 8.0.1024000000.1, 8.0.53824000000.4, 8.0.64000000.2, 8.8.861184000000.3, \(\Q(\zeta_{40})^+\), 8.0.861184000000.17, 8.0.53824000000.11, 8.0.861184000000.40, 8.0.53824000000.2, 8.0.861184000000.32, 8.0.3364000000.3, \(\Q(\zeta_{20})\), 8.0.1024000000.2, 8.0.53824000000.6, 8.0.13140625.1, 8.8.3364000000.1, 8.8.53824000000.1, 8.0.861184000000.28, 8.0.3364000000.5, 8.0.3364000000.2, 8.0.861184000000.25, 16.0.1186620610969600000000.3, 16.0.741637881856000000000000.9, 16.0.741637881856000000000000.3, 16.0.2897022976000000000000.1, 16.16.741637881856000000000000.1, 16.0.741637881856000000000000.8, \(\Q(\zeta_{40})\), 16.0.741637881856000000000000.10, 16.0.741637881856000000000000.5, 16.0.741637881856000000000000.4, 16.0.2897022976000000000000.3, 16.0.741637881856000000000000.2, 16.0.741637881856000000000000.7, 16.0.741637881856000000000000.6, 16.0.11316496000000000000.1

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 32 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 ${\href{/padicField/3.4.0.1}{4} }^{8}$ R ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ R ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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$$4$$4$$32$
Deg $16$$4$$4$$32$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(29\) Copy content Toggle raw display 29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$