LMFDB - Number field 32.0.87993561227221187133696000000000000000000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 5*x^30 + 19*x^28 - 70*x^26 + 215*x^24 - 590*x^22 + 1466*x^20 - 3340*x^18 + 7009*x^16 - 13360*x^14 + 23456*x^12 - 37760*x^10 + 55040*x^8 - 71680*x^6 + 77824*x^4 - 81920*x^2 + 65536)

gp: K = bnfinit(y^32 - 5*y^30 + 19*y^28 - 70*y^26 + 215*y^24 - 590*y^22 + 1466*y^20 - 3340*y^18 + 7009*y^16 - 13360*y^14 + 23456*y^12 - 37760*y^10 + 55040*y^8 - 71680*y^6 + 77824*y^4 - 81920*y^2 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 5*x^30 + 19*x^28 - 70*x^26 + 215*x^24 - 590*x^22 + 1466*x^20 - 3340*x^18 + 7009*x^16 - 13360*x^14 + 23456*x^12 - 37760*x^10 + 55040*x^8 - 71680*x^6 + 77824*x^4 - 81920*x^2 + 65536);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 5*x^30 + 19*x^28 - 70*x^26 + 215*x^24 - 590*x^22 + 1466*x^20 - 3340*x^18 + 7009*x^16 - 13360*x^14 + 23456*x^12 - 37760*x^10 + 55040*x^8 - 71680*x^6 + 77824*x^4 - 81920*x^2 + 65536)

$ x^{32} - 5 x^{30} + 19 x^{28} - 70 x^{26} + 215 x^{24} - 590 x^{22} + 1466 x^{20} - 3340 x^{18} + \cdots + 65536 $ x^32 - 5*x^30 + 19*x^28 - 70*x^26 + 215*x^24 - 590*x^22 + 1466*x^20 - 3340*x^18 + 7009*x^16 - 13360*x^14 + 23456*x^12 - 37760*x^10 + 55040*x^8 - 71680*x^6 + 77824*x^4 - 81920*x^2 + 65536

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:	$87993561227221187133696000000000000000000000000$ 87993561227221187133696000000000000000000000000 $\medspace = 2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 41^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$29.31$	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:	$2\cdot 3^{1/2}5^{3/4}41^{1/2}\approx 74.16688765648375$
Ramified primes:	$2$, $3$, $5$, $41$ 2, 3, 5, 41	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{14}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{17}-\frac{1}{6}a^{15}-\frac{1}{2}a^{13}-\frac{1}{3}a^{11}-\frac{1}{6}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{6}a$, $\frac{1}{12}a^{18}-\frac{1}{12}a^{16}+\frac{1}{4}a^{14}-\frac{1}{6}a^{12}-\frac{1}{12}a^{10}-\frac{1}{6}a^{8}-\frac{1}{2}a^{6}-\frac{1}{3}a^{4}+\frac{1}{12}a^{2}$, $\frac{1}{24}a^{19}-\frac{1}{24}a^{17}-\frac{3}{8}a^{15}-\frac{1}{12}a^{13}-\frac{1}{24}a^{11}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}-\frac{1}{6}a^{5}+\frac{1}{24}a^{3}-\frac{1}{2}a$, $\frac{1}{48}a^{20}-\frac{1}{48}a^{18}-\frac{1}{48}a^{16}-\frac{5}{24}a^{14}-\frac{1}{48}a^{12}-\frac{3}{8}a^{10}-\frac{7}{24}a^{8}-\frac{5}{12}a^{6}+\frac{1}{48}a^{4}+\frac{1}{12}a^{2}-\frac{1}{3}$, $\frac{1}{96}a^{21}-\frac{1}{96}a^{19}-\frac{1}{96}a^{17}+\frac{19}{48}a^{15}-\frac{1}{96}a^{13}-\frac{3}{16}a^{11}+\frac{17}{48}a^{9}+\frac{7}{24}a^{7}-\frac{47}{96}a^{5}-\frac{11}{24}a^{3}+\frac{1}{3}a$, $\frac{1}{192}a^{22}-\frac{1}{192}a^{20}-\frac{1}{192}a^{18}+\frac{1}{32}a^{16}-\frac{65}{192}a^{14}+\frac{13}{32}a^{12}+\frac{1}{96}a^{10}-\frac{3}{16}a^{8}-\frac{79}{192}a^{6}-\frac{11}{48}a^{4}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{384}a^{23}-\frac{1}{384}a^{21}-\frac{1}{384}a^{19}+\frac{1}{64}a^{17}-\frac{65}{384}a^{15}-\frac{19}{64}a^{13}+\frac{1}{192}a^{11}-\frac{3}{32}a^{9}-\frac{79}{384}a^{7}-\frac{11}{96}a^{5}-\frac{1}{12}a^{3}-\frac{1}{3}a$, $\frac{1}{2304}a^{24}+\frac{1}{768}a^{22}-\frac{7}{768}a^{20}+\frac{41}{1152}a^{18}-\frac{115}{768}a^{16}-\frac{25}{384}a^{14}-\frac{475}{1152}a^{12}-\frac{5}{192}a^{10}-\frac{341}{768}a^{8}-\frac{133}{288}a^{6}+\frac{1}{8}a^{4}+\frac{1}{6}a^{2}+\frac{1}{9}$, $\frac{1}{4608}a^{25}+\frac{1}{1536}a^{23}-\frac{7}{1536}a^{21}+\frac{41}{2304}a^{19}-\frac{115}{1536}a^{17}+\frac{359}{768}a^{15}+\frac{677}{2304}a^{13}+\frac{187}{384}a^{11}+\frac{427}{1536}a^{9}+\frac{155}{576}a^{7}-\frac{7}{16}a^{5}+\frac{1}{12}a^{3}+\frac{1}{18}a$, $\frac{1}{9216}a^{26}-\frac{1}{9216}a^{24}+\frac{5}{3072}a^{22}-\frac{37}{4608}a^{20}+\frac{239}{9216}a^{18}+\frac{29}{1536}a^{16}+\frac{761}{4608}a^{14}-\frac{193}{2304}a^{12}-\frac{375}{1024}a^{10}+\frac{805}{2304}a^{8}-\frac{49}{576}a^{6}+\frac{1}{3}a^{4}-\frac{17}{36}a^{2}+\frac{2}{9}$, $\frac{1}{18432}a^{27}-\frac{1}{18432}a^{25}+\frac{5}{6144}a^{23}-\frac{37}{9216}a^{21}+\frac{239}{18432}a^{19}+\frac{29}{3072}a^{17}+\frac{761}{9216}a^{15}+\frac{2111}{4608}a^{13}-\frac{375}{2048}a^{11}+\frac{805}{4608}a^{9}+\frac{527}{1152}a^{7}-\frac{1}{3}a^{5}+\frac{19}{72}a^{3}+\frac{1}{9}a$, $\frac{1}{700416}a^{28}+\frac{1}{233472}a^{26}-\frac{37}{700416}a^{24}+\frac{305}{350208}a^{22}-\frac{321}{77824}a^{20}-\frac{12539}{350208}a^{18}+\frac{43949}{350208}a^{16}+\frac{3283}{58368}a^{14}+\frac{243425}{700416}a^{12}+\frac{20771}{87552}a^{10}+\frac{2345}{4864}a^{8}+\frac{3257}{10944}a^{6}+\frac{973}{2736}a^{4}+\frac{23}{228}a^{2}+\frac{23}{171}$, $\frac{1}{1400832}a^{29}+\frac{1}{466944}a^{27}-\frac{37}{1400832}a^{25}+\frac{305}{700416}a^{23}-\frac{321}{155648}a^{21}-\frac{12539}{700416}a^{19}+\frac{43949}{700416}a^{17}-\frac{55085}{116736}a^{15}-\frac{456991}{1400832}a^{13}+\frac{20771}{175104}a^{11}+\frac{2345}{9728}a^{9}-\frac{7687}{21888}a^{7}-\frac{1763}{5472}a^{5}-\frac{205}{456}a^{3}+\frac{23}{342}a$, $\frac{1}{669597696}a^{30}+\frac{157}{223199232}a^{28}-\frac{20521}{669597696}a^{26}-\frac{6635}{111599616}a^{24}+\frac{576949}{223199232}a^{22}+\frac{2497963}{334798848}a^{20}+\frac{942107}{111599616}a^{18}+\frac{117923}{2936832}a^{16}+\frac{50206769}{669597696}a^{14}-\frac{2330135}{55799808}a^{12}-\frac{2291539}{13949952}a^{10}-\frac{1458121}{10462464}a^{8}+\frac{268625}{871872}a^{6}-\frac{46105}{217968}a^{4}+\frac{58541}{163476}a^{2}-\frac{1432}{40869}$, $\frac{1}{1339195392}a^{31}+\frac{157}{446398464}a^{29}-\frac{20521}{1339195392}a^{27}-\frac{6635}{223199232}a^{25}+\frac{576949}{446398464}a^{23}+\frac{2497963}{669597696}a^{21}+\frac{942107}{223199232}a^{19}+\frac{117923}{5873664}a^{17}+\frac{50206769}{1339195392}a^{15}-\frac{2330135}{111599616}a^{13}-\frac{2291539}{27899904}a^{11}+\frac{9004343}{20924928}a^{9}-\frac{603247}{1743744}a^{7}+\frac{171863}{435936}a^{5}+\frac{58541}{326952}a^{3}-\frac{716}{40869}a$ 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, 1/3*a^16 - 1/3*a^14 + 1/3*a^10 - 1/3*a^8 + 1/3*a^6 - 1/3*a^2 + 1/3, 1/6*a^17 - 1/6*a^15 - 1/2*a^13 - 1/3*a^11 - 1/6*a^9 - 1/3*a^7 + 1/3*a^3 + 1/6*a, 1/12*a^18 - 1/12*a^16 + 1/4*a^14 - 1/6*a^12 - 1/12*a^10 - 1/6*a^8 - 1/2*a^6 - 1/3*a^4 + 1/12*a^2, 1/24*a^19 - 1/24*a^17 - 3/8*a^15 - 1/12*a^13 - 1/24*a^11 - 1/12*a^9 - 1/4*a^7 - 1/6*a^5 + 1/24*a^3 - 1/2*a, 1/48*a^20 - 1/48*a^18 - 1/48*a^16 - 5/24*a^14 - 1/48*a^12 - 3/8*a^10 - 7/24*a^8 - 5/12*a^6 + 1/48*a^4 + 1/12*a^2 - 1/3, 1/96*a^21 - 1/96*a^19 - 1/96*a^17 + 19/48*a^15 - 1/96*a^13 - 3/16*a^11 + 17/48*a^9 + 7/24*a^7 - 47/96*a^5 - 11/24*a^3 + 1/3*a, 1/192*a^22 - 1/192*a^20 - 1/192*a^18 + 1/32*a^16 - 65/192*a^14 + 13/32*a^12 + 1/96*a^10 - 3/16*a^8 - 79/192*a^6 - 11/48*a^4 - 1/6*a^2 + 1/3, 1/384*a^23 - 1/384*a^21 - 1/384*a^19 + 1/64*a^17 - 65/384*a^15 - 19/64*a^13 + 1/192*a^11 - 3/32*a^9 - 79/384*a^7 - 11/96*a^5 - 1/12*a^3 - 1/3*a, 1/2304*a^24 + 1/768*a^22 - 7/768*a^20 + 41/1152*a^18 - 115/768*a^16 - 25/384*a^14 - 475/1152*a^12 - 5/192*a^10 - 341/768*a^8 - 133/288*a^6 + 1/8*a^4 + 1/6*a^2 + 1/9, 1/4608*a^25 + 1/1536*a^23 - 7/1536*a^21 + 41/2304*a^19 - 115/1536*a^17 + 359/768*a^15 + 677/2304*a^13 + 187/384*a^11 + 427/1536*a^9 + 155/576*a^7 - 7/16*a^5 + 1/12*a^3 + 1/18*a, 1/9216*a^26 - 1/9216*a^24 + 5/3072*a^22 - 37/4608*a^20 + 239/9216*a^18 + 29/1536*a^16 + 761/4608*a^14 - 193/2304*a^12 - 375/1024*a^10 + 805/2304*a^8 - 49/576*a^6 + 1/3*a^4 - 17/36*a^2 + 2/9, 1/18432*a^27 - 1/18432*a^25 + 5/6144*a^23 - 37/9216*a^21 + 239/18432*a^19 + 29/3072*a^17 + 761/9216*a^15 + 2111/4608*a^13 - 375/2048*a^11 + 805/4608*a^9 + 527/1152*a^7 - 1/3*a^5 + 19/72*a^3 + 1/9*a, 1/700416*a^28 + 1/233472*a^26 - 37/700416*a^24 + 305/350208*a^22 - 321/77824*a^20 - 12539/350208*a^18 + 43949/350208*a^16 + 3283/58368*a^14 + 243425/700416*a^12 + 20771/87552*a^10 + 2345/4864*a^8 + 3257/10944*a^6 + 973/2736*a^4 + 23/228*a^2 + 23/171, 1/1400832*a^29 + 1/466944*a^27 - 37/1400832*a^25 + 305/700416*a^23 - 321/155648*a^21 - 12539/700416*a^19 + 43949/700416*a^17 - 55085/116736*a^15 - 456991/1400832*a^13 + 20771/175104*a^11 + 2345/9728*a^9 - 7687/21888*a^7 - 1763/5472*a^5 - 205/456*a^3 + 23/342*a, 1/669597696*a^30 + 157/223199232*a^28 - 20521/669597696*a^26 - 6635/111599616*a^24 + 576949/223199232*a^22 + 2497963/334798848*a^20 + 942107/111599616*a^18 + 117923/2936832*a^16 + 50206769/669597696*a^14 - 2330135/55799808*a^12 - 2291539/13949952*a^10 - 1458121/10462464*a^8 + 268625/871872*a^6 - 46105/217968*a^4 + 58541/163476*a^2 - 1432/40869, 1/1339195392*a^31 + 157/446398464*a^29 - 20521/1339195392*a^27 - 6635/223199232*a^25 + 576949/446398464*a^23 + 2497963/669597696*a^21 + 942107/223199232*a^19 + 117923/5873664*a^17 + 50206769/1339195392*a^15 - 2330135/111599616*a^13 - 2291539/27899904*a^11 + 9004343/20924928*a^9 - 603247/1743744*a^7 + 171863/435936*a^5 + 58541/326952*a^3 - 716/40869*a

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

$C_{4}\times C_{4}$, which has order $16$ (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{113743}{669597696} a^{31} + \frac{930041}{669597696} a^{29} - \frac{374675}{74399744} a^{27} + \frac{142897}{8810496} a^{25} - \frac{30438373}{669597696} a^{23} + \frac{6160595}{55799808} a^{21} - \frac{4125595}{17620992} a^{19} + \frac{18279485}{41849856} a^{17} - \frac{160684685}{223199232} a^{15} + \frac{17595125}{17620992} a^{13} - \frac{89290565}{83699712} a^{11} + \frac{2441033}{3487488} a^{9} + \frac{86357}{275328} a^{7} - \frac{2488325}{1307808} a^{5} + \frac{506723}{108984} a^{3} - \frac{575189}{81738} a $ -(113743)/(669597696)a^(31) + (930041)/(669597696)a^(29) - (374675)/(74399744)a^(27) + (142897)/(8810496)a^(25) - (30438373)/(669597696)a^(23) + (6160595)/(55799808)a^(21) - (4125595)/(17620992)a^(19) + (18279485)/(41849856)a^(17) - (160684685)/(223199232)a^(15) + (17595125)/(17620992)a^(13) - (89290565)/(83699712)a^(11) + (2441033)/(3487488)a^(9) + (86357)/(275328)a^(7) - (2488325)/(1307808)a^(5) + (506723)/(108984)a^(3) - (575189)/(81738)a (order $60$)	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:	$\frac{61}{73728}a^{30}-\frac{283}{73728}a^{28}+\frac{785}{73728}a^{26}-\frac{505}{18432}a^{24}+\frac{4127}{73728}a^{22}-\frac{1435}{18432}a^{20}+\frac{307}{36864}a^{18}+\frac{1625}{4608}a^{16}-\frac{102715}{73728}a^{14}+\frac{141859}{36864}a^{12}-\frac{78785}{9216}a^{10}+\frac{18143}{1152}a^{8}-\frac{14141}{576}a^{6}+\frac{4735}{144}a^{4}-\frac{1451}{36}a^{2}+\frac{260}{9}$, $\frac{434285}{669597696}a^{30}-\frac{1685821}{669597696}a^{28}+\frac{4563475}{669597696}a^{26}-\frac{1851235}{111599616}a^{24}+\frac{20057795}{669597696}a^{22}-\frac{9391501}{334798848}a^{20}-\frac{6719689}{111599616}a^{18}+\frac{3357833}{8810496}a^{16}-\frac{815089283}{669597696}a^{14}+\frac{171104303}{55799808}a^{12}-\frac{267220387}{41849856}a^{10}+\frac{118148851}{10462464}a^{8}-\frac{2449631}{145312}a^{6}+\frac{7077277}{326952}a^{4}-\frac{4232759}{163476}a^{2}+\frac{593242}{40869}$, $\frac{55343}{223199232}a^{30}-\frac{1415065}{669597696}a^{28}+\frac{2406133}{223199232}a^{26}-\frac{13645795}{334798848}a^{24}+\frac{88133195}{669597696}a^{22}-\frac{13937639}{37199872}a^{20}+\frac{315709577}{334798848}a^{18}-\frac{355555771}{167399424}a^{16}+\frac{50661973}{11747328}a^{14}-\frac{661489825}{83699712}a^{12}+\frac{544103021}{41849856}a^{10}-\frac{22011335}{1162496}a^{8}+\frac{63451033}{2615616}a^{6}-\frac{17340403}{653904}a^{4}+\frac{620701}{27246}a^{2}-\frac{519977}{40869}$, $\frac{127567}{669597696}a^{31}-\frac{516305}{669597696}a^{29}+\frac{134739}{74399744}a^{27}-\frac{31933}{8810496}a^{25}+\frac{2232685}{669597696}a^{23}+\frac{693779}{55799808}a^{21}-\frac{1408025}{17620992}a^{19}+\frac{11776795}{41849856}a^{17}-\frac{169961315}{223199232}a^{15}+\frac{30872695}{17620992}a^{13}-\frac{146130053}{41849856}a^{11}+\frac{41965541}{6974976}a^{9}-\frac{2466665}{275328}a^{7}+\frac{14984921}{1307808}a^{5}-\frac{696091}{54492}a^{3}+\frac{352885}{40869}a-1$, $\frac{2255}{27899904}a^{31}-\frac{382433}{669597696}a^{30}-\frac{179143}{334798848}a^{29}+\frac{2394533}{669597696}a^{28}+\frac{726875}{334798848}a^{27}-\frac{2641313}{223199232}a^{26}-\frac{279289}{37199872}a^{25}+\frac{11916155}{334798848}a^{24}+\frac{3755287}{167399424}a^{23}-\frac{61536583}{669597696}a^{22}-\frac{19615645}{334798848}a^{21}+\frac{22297981}{111599616}a^{20}+\frac{7534027}{55799808}a^{19}-\frac{121500229}{334798848}a^{18}-\frac{46616015}{167399424}a^{17}+\frac{86992523}{167399424}a^{16}+\frac{21609055}{41849856}a^{15}-\frac{116128139}{223199232}a^{14}-\frac{94975969}{111599616}a^{13}-\frac{3506885}{41849856}a^{12}+\frac{6399365}{5231232}a^{11}+\frac{83573735}{41849856}a^{10}-\frac{62084}{40869}a^{9}-\frac{3271465}{581248}a^{8}+\frac{431325}{290624}a^{7}+\frac{28923913}{2615616}a^{6}-\frac{99625}{81738}a^{5}-\frac{2784775}{163476}a^{4}+\frac{6082}{40869}a^{3}+\frac{1310591}{54492}a^{2}+\frac{22687}{27246}a-\frac{316697}{13623}$, $\frac{110905}{446398464}a^{31}+\frac{106583}{223199232}a^{30}-\frac{161107}{148799488}a^{29}-\frac{265691}{223199232}a^{28}+\frac{5547125}{1339195392}a^{27}+\frac{115543}{669597696}a^{26}-\frac{9136655}{669597696}a^{25}+\frac{2703661}{334798848}a^{24}+\frac{5747021}{148799488}a^{23}-\frac{3620405}{74399744}a^{22}-\frac{66204755}{669597696}a^{21}+\frac{64323737}{334798848}a^{20}+\frac{151494331}{669597696}a^{19}-\frac{204216695}{334798848}a^{18}-\frac{53019985}{111599616}a^{17}+\frac{90276311}{55799808}a^{16}+\frac{1221020027}{1339195392}a^{15}-\frac{2490957755}{669597696}a^{14}-\frac{526261993}{334798848}a^{13}+\frac{642994291}{83699712}a^{12}+\frac{8678395}{3487488}a^{11}-\frac{48901553}{3487488}a^{10}-\frac{143456}{40869}a^{9}+\frac{236145985}{10462464}a^{8}+\frac{11947045}{2615616}a^{7}-\frac{41396765}{1307808}a^{6}-\frac{27647}{5736}a^{5}+\frac{4200301}{108984}a^{4}+\frac{143293}{40869}a^{3}-\frac{3348637}{81738}a^{2}-\frac{287941}{81738}a+\frac{994652}{40869}$, $\frac{23601}{74399744}a^{31}-\frac{54789}{74399744}a^{29}+\frac{169795}{669597696}a^{27}+\frac{1386353}{334798848}a^{25}-\frac{6108787}{223199232}a^{23}+\frac{36629585}{334798848}a^{21}-\frac{116271319}{334798848}a^{19}+\frac{901675}{978944}a^{17}-\frac{1418296055}{669597696}a^{15}+\frac{91365413}{20924928}a^{13}-\frac{221264615}{27899904}a^{11}+\frac{132842971}{10462464}a^{9}-\frac{46326431}{2615616}a^{7}+\frac{9339065}{435936}a^{5}-\frac{7440007}{326952}a^{3}+\frac{504166}{40869}a+1$, $\frac{7793}{37199872}a^{30}-\frac{16789}{37199872}a^{28}-\frac{31055}{111599616}a^{26}+\frac{780025}{167399424}a^{24}-\frac{967601}{37199872}a^{22}+\frac{5423315}{55799808}a^{20}-\frac{49347239}{167399424}a^{18}+\frac{377555}{489472}a^{16}-\frac{194937725}{111599616}a^{14}+\frac{74331437}{20924928}a^{12}-\frac{29691085}{4649984}a^{10}+\frac{17658697}{1743744}a^{8}-\frac{36803939}{2615616}a^{6}+\frac{1220115}{72656}a^{4}-\frac{958381}{54492}a^{2}+\frac{402916}{40869}$, $\frac{92519}{111599616}a^{30}-\frac{2209145}{334798848}a^{28}+\frac{10357727}{334798848}a^{26}-\frac{6375181}{55799808}a^{24}+\frac{120252835}{334798848}a^{22}-\frac{166204475}{167399424}a^{20}+\frac{135819131}{55799808}a^{18}-\frac{446998631}{83699712}a^{16}+\frac{186605687}{17620992}a^{14}-\frac{21956855}{1162496}a^{12}+\frac{315989207}{10462464}a^{10}-\frac{223699489}{5231232}a^{8}+\frac{46334887}{871872}a^{6}-\frac{9219241}{163476}a^{4}+\frac{3693131}{81738}a^{2}-\frac{293627}{13623}$, $\frac{833}{23494656}a^{31}-\frac{587141}{1339195392}a^{29}+\frac{385851}{148799488}a^{27}-\frac{6770927}{669597696}a^{25}+\frac{44319679}{1339195392}a^{23}-\frac{21501017}{223199232}a^{21}+\frac{162510949}{669597696}a^{19}-\frac{183009143}{334798848}a^{17}+\frac{495875443}{446398464}a^{15}-\frac{335062403}{167399424}a^{13}+\frac{271461943}{83699712}a^{11}-\frac{10772607}{2324992}a^{9}+\frac{14716789}{2615616}a^{7}-\frac{3914353}{653904}a^{5}+\frac{509045}{108984}a^{3}-\frac{61742}{40869}a$, $\frac{5243}{111599616}a^{31}-\frac{243943}{334798848}a^{29}+\frac{351791}{111599616}a^{27}-\frac{933199}{83699712}a^{25}+\frac{11722067}{334798848}a^{23}-\frac{2646557}{27899904}a^{21}+\frac{37079299}{167399424}a^{19}-\frac{9583081}{20924928}a^{17}+\frac{31822177}{37199872}a^{15}-\frac{234249377}{167399424}a^{13}+\frac{40499495}{20924928}a^{11}-\frac{16026269}{6974976}a^{9}+\frac{11083829}{5231232}a^{7}-\frac{614195}{653904}a^{5}-\frac{30073}{36328}a^{3}+\frac{147283}{40869}a$, $\frac{48385}{83699712}a^{31}-\frac{302549}{669597696}a^{30}-\frac{76249}{20924928}a^{29}+\frac{387005}{669597696}a^{28}+\frac{1060867}{83699712}a^{27}+\frac{2106893}{669597696}a^{26}-\frac{43375}{1101312}a^{25}-\frac{2281025}{111599616}a^{24}+\frac{1101283}{10462464}a^{23}+\frac{59363045}{669597696}a^{22}-\frac{20321501}{83699712}a^{21}-\frac{102152615}{334798848}a^{20}+\frac{2117285}{4405248}a^{19}+\frac{98744465}{111599616}a^{18}-\frac{527579}{653904}a^{17}-\frac{371284735}{167399424}a^{16}+\frac{95689735}{83699712}a^{15}+\frac{3285135323}{669597696}a^{14}-\frac{5015065}{4405248}a^{13}-\frac{544668565}{55799808}a^{12}+\frac{14348741}{83699712}a^{11}+\frac{361452463}{20924928}a^{10}+\frac{23349695}{10462464}a^{9}-\frac{8841059}{326952}a^{8}-\frac{436063}{68832}a^{7}+\frac{10755705}{290624}a^{6}+\frac{14820335}{1307808}a^{5}-\frac{1515865}{34416}a^{4}-\frac{2984905}{163476}a^{3}+\frac{7426097}{163476}a^{2}+\frac{771053}{40869}a-\frac{1032430}{40869}$, $\frac{37909}{167399424}a^{31}+\frac{112637}{111599616}a^{30}-\frac{7487}{41849856}a^{29}-\frac{2243965}{334798848}a^{28}-\frac{93491}{41849856}a^{27}+\frac{3219901}{111599616}a^{26}+\frac{746425}{55799808}a^{25}-\frac{4231139}{41849856}a^{24}-\frac{9469769}{167399424}a^{23}+\frac{102178877}{334798848}a^{22}+\frac{32123761}{167399424}a^{21}-\frac{11342611}{13949952}a^{20}-\frac{15454951}{27899904}a^{19}+\frac{321761785}{167399424}a^{18}+\frac{115814201}{83699712}a^{17}-\frac{170762075}{41849856}a^{16}-\frac{510823307}{167399424}a^{15}+\frac{875134373}{111599616}a^{14}+\frac{339358951}{55799808}a^{13}-\frac{2265417437}{167399424}a^{12}-\frac{900722725}{83699712}a^{11}+\frac{436386533}{20924928}a^{10}+\frac{175848859}{10462464}a^{9}-\frac{33033047}{1162496}a^{8}-\frac{13382337}{581248}a^{7}+\frac{44114819}{1307808}a^{6}+\frac{35533505}{1307808}a^{5}-\frac{22161593}{653904}a^{4}-\frac{8951479}{326952}a^{3}+\frac{423431}{18164}a^{2}+\frac{1252561}{81738}a-\frac{342866}{40869}$, $\frac{687083}{669597696}a^{31}-\frac{40499}{41849856}a^{30}-\frac{1290263}{223199232}a^{29}+\frac{109601}{20924928}a^{28}+\frac{13282951}{669597696}a^{27}-\frac{81417}{4649984}a^{26}-\frac{179791}{2936832}a^{25}+\frac{547397}{10462464}a^{24}+\frac{36321899}{223199232}a^{23}-\frac{2822359}{20924928}a^{22}-\frac{62269607}{167399424}a^{21}+\frac{1383697}{4649984}a^{20}+\frac{4256017}{5873664}a^{19}-\frac{23117263}{41849856}a^{18}-\frac{5578669}{4649984}a^{17}+\frac{8867449}{10462464}a^{16}+\frac{1101866515}{669597696}a^{15}-\frac{14110291}{13949952}a^{14}-\frac{2887453}{1957888}a^{13}+\frac{20655671}{41849856}a^{12}-\frac{3409519}{13949952}a^{11}+\frac{64243399}{41849856}a^{10}+\frac{706039}{163476}a^{9}-\frac{19791203}{3487488}a^{8}-\frac{1013173}{91776}a^{7}+\frac{31182233}{2615616}a^{6}+\frac{523175}{27246}a^{5}-\frac{3137663}{163476}a^{4}-\frac{9914695}{326952}a^{3}+\frac{786757}{27246}a^{2}+\frac{1118197}{40869}a-\frac{1033307}{40869}$, $\frac{419777}{446398464}a^{31}+\frac{178859}{167399424}a^{30}-\frac{340313}{70483968}a^{29}-\frac{1485277}{167399424}a^{28}+\frac{7499543}{446398464}a^{27}+\frac{6428147}{167399424}a^{26}-\frac{34911299}{669597696}a^{25}-\frac{2848447}{20924928}a^{24}+\frac{185677309}{1339195392}a^{23}+\frac{69609533}{167399424}a^{22}-\frac{71398213}{223199232}a^{21}-\frac{46607947}{41849856}a^{20}+\frac{428614339}{669597696}a^{19}+\frac{220907735}{83699712}a^{18}-\frac{371084111}{334798848}a^{17}-\frac{117007481}{20924928}a^{16}+\frac{738852529}{446398464}a^{15}+\frac{1790753435}{167399424}a^{14}-\frac{639955783}{334798848}a^{13}-\frac{1533241363}{83699712}a^{12}+\frac{103669951}{83699712}a^{11}+\frac{1164296249}{41849856}a^{10}+\frac{3260797}{3487488}a^{9}-\frac{388108465}{10462464}a^{8}-\frac{12519719}{2615616}a^{7}+\frac{27802867}{653904}a^{6}+\frac{6592223}{653904}a^{5}-\frac{26673571}{653904}a^{4}-\frac{1989065}{108984}a^{3}+\frac{982745}{40869}a^{2}+\frac{611641}{40869}a+\frac{106334}{40869}$ 61/73728a^30 - 283/73728a^28 + 785/73728a^26 - 505/18432a^24 + 4127/73728a^22 - 1435/18432a^20 + 307/36864a^18 + 1625/4608a^16 - 102715/73728a^14 + 141859/36864a^12 - 78785/9216a^10 + 18143/1152a^8 - 14141/576a^6 + 4735/144a^4 - 1451/36a^2 + 260/9, 434285/669597696a^30 - 1685821/669597696a^28 + 4563475/669597696a^26 - 1851235/111599616a^24 + 20057795/669597696a^22 - 9391501/334798848a^20 - 6719689/111599616a^18 + 3357833/8810496a^16 - 815089283/669597696a^14 + 171104303/55799808a^12 - 267220387/41849856a^10 + 118148851/10462464a^8 - 2449631/145312a^6 + 7077277/326952a^4 - 4232759/163476a^2 + 593242/40869, 55343/223199232a^30 - 1415065/669597696a^28 + 2406133/223199232a^26 - 13645795/334798848a^24 + 88133195/669597696a^22 - 13937639/37199872a^20 + 315709577/334798848a^18 - 355555771/167399424a^16 + 50661973/11747328a^14 - 661489825/83699712a^12 + 544103021/41849856a^10 - 22011335/1162496a^8 + 63451033/2615616a^6 - 17340403/653904a^4 + 620701/27246a^2 - 519977/40869, 127567/669597696a^31 - 516305/669597696a^29 + 134739/74399744a^27 - 31933/8810496a^25 + 2232685/669597696a^23 + 693779/55799808a^21 - 1408025/17620992a^19 + 11776795/41849856a^17 - 169961315/223199232a^15 + 30872695/17620992a^13 - 146130053/41849856a^11 + 41965541/6974976a^9 - 2466665/275328a^7 + 14984921/1307808a^5 - 696091/54492a^3 + 352885/40869a - 1, 2255/27899904a^31 - 382433/669597696a^30 - 179143/334798848a^29 + 2394533/669597696a^28 + 726875/334798848a^27 - 2641313/223199232a^26 - 279289/37199872a^25 + 11916155/334798848a^24 + 3755287/167399424a^23 - 61536583/669597696a^22 - 19615645/334798848a^21 + 22297981/111599616a^20 + 7534027/55799808a^19 - 121500229/334798848a^18 - 46616015/167399424a^17 + 86992523/167399424a^16 + 21609055/41849856a^15 - 116128139/223199232a^14 - 94975969/111599616a^13 - 3506885/41849856a^12 + 6399365/5231232a^11 + 83573735/41849856a^10 - 62084/40869a^9 - 3271465/581248a^8 + 431325/290624a^7 + 28923913/2615616a^6 - 99625/81738a^5 - 2784775/163476a^4 + 6082/40869a^3 + 1310591/54492a^2 + 22687/27246a - 316697/13623, 110905/446398464a^31 + 106583/223199232a^30 - 161107/148799488a^29 - 265691/223199232a^28 + 5547125/1339195392a^27 + 115543/669597696a^26 - 9136655/669597696a^25 + 2703661/334798848a^24 + 5747021/148799488a^23 - 3620405/74399744a^22 - 66204755/669597696a^21 + 64323737/334798848a^20 + 151494331/669597696a^19 - 204216695/334798848a^18 - 53019985/111599616a^17 + 90276311/55799808a^16 + 1221020027/1339195392a^15 - 2490957755/669597696a^14 - 526261993/334798848a^13 + 642994291/83699712a^12 + 8678395/3487488a^11 - 48901553/3487488a^10 - 143456/40869a^9 + 236145985/10462464a^8 + 11947045/2615616a^7 - 41396765/1307808a^6 - 27647/5736a^5 + 4200301/108984a^4 + 143293/40869a^3 - 3348637/81738a^2 - 287941/81738a + 994652/40869, 23601/74399744a^31 - 54789/74399744a^29 + 169795/669597696a^27 + 1386353/334798848a^25 - 6108787/223199232a^23 + 36629585/334798848a^21 - 116271319/334798848a^19 + 901675/978944a^17 - 1418296055/669597696a^15 + 91365413/20924928a^13 - 221264615/27899904a^11 + 132842971/10462464a^9 - 46326431/2615616a^7 + 9339065/435936a^5 - 7440007/326952a^3 + 504166/40869a + 1, 7793/37199872a^30 - 16789/37199872a^28 - 31055/111599616a^26 + 780025/167399424a^24 - 967601/37199872a^22 + 5423315/55799808a^20 - 49347239/167399424a^18 + 377555/489472a^16 - 194937725/111599616a^14 + 74331437/20924928a^12 - 29691085/4649984a^10 + 17658697/1743744a^8 - 36803939/2615616a^6 + 1220115/72656a^4 - 958381/54492a^2 + 402916/40869, 92519/111599616a^30 - 2209145/334798848a^28 + 10357727/334798848a^26 - 6375181/55799808a^24 + 120252835/334798848a^22 - 166204475/167399424a^20 + 135819131/55799808a^18 - 446998631/83699712a^16 + 186605687/17620992a^14 - 21956855/1162496a^12 + 315989207/10462464a^10 - 223699489/5231232a^8 + 46334887/871872a^6 - 9219241/163476a^4 + 3693131/81738a^2 - 293627/13623, 833/23494656a^31 - 587141/1339195392a^29 + 385851/148799488a^27 - 6770927/669597696a^25 + 44319679/1339195392a^23 - 21501017/223199232a^21 + 162510949/669597696a^19 - 183009143/334798848a^17 + 495875443/446398464a^15 - 335062403/167399424a^13 + 271461943/83699712a^11 - 10772607/2324992a^9 + 14716789/2615616a^7 - 3914353/653904a^5 + 509045/108984a^3 - 61742/40869a, 5243/111599616a^31 - 243943/334798848a^29 + 351791/111599616a^27 - 933199/83699712a^25 + 11722067/334798848a^23 - 2646557/27899904a^21 + 37079299/167399424a^19 - 9583081/20924928a^17 + 31822177/37199872a^15 - 234249377/167399424a^13 + 40499495/20924928a^11 - 16026269/6974976a^9 + 11083829/5231232a^7 - 614195/653904a^5 - 30073/36328a^3 + 147283/40869a, 48385/83699712a^31 - 302549/669597696a^30 - 76249/20924928a^29 + 387005/669597696a^28 + 1060867/83699712a^27 + 2106893/669597696a^26 - 43375/1101312a^25 - 2281025/111599616a^24 + 1101283/10462464a^23 + 59363045/669597696a^22 - 20321501/83699712a^21 - 102152615/334798848a^20 + 2117285/4405248a^19 + 98744465/111599616a^18 - 527579/653904a^17 - 371284735/167399424a^16 + 95689735/83699712a^15 + 3285135323/669597696a^14 - 5015065/4405248a^13 - 544668565/55799808a^12 + 14348741/83699712a^11 + 361452463/20924928a^10 + 23349695/10462464a^9 - 8841059/326952a^8 - 436063/68832a^7 + 10755705/290624a^6 + 14820335/1307808a^5 - 1515865/34416a^4 - 2984905/163476a^3 + 7426097/163476a^2 + 771053/40869a - 1032430/40869, 37909/167399424a^31 + 112637/111599616a^30 - 7487/41849856a^29 - 2243965/334798848a^28 - 93491/41849856a^27 + 3219901/111599616a^26 + 746425/55799808a^25 - 4231139/41849856a^24 - 9469769/167399424a^23 + 102178877/334798848a^22 + 32123761/167399424a^21 - 11342611/13949952a^20 - 15454951/27899904a^19 + 321761785/167399424a^18 + 115814201/83699712a^17 - 170762075/41849856a^16 - 510823307/167399424a^15 + 875134373/111599616a^14 + 339358951/55799808a^13 - 2265417437/167399424a^12 - 900722725/83699712a^11 + 436386533/20924928a^10 + 175848859/10462464a^9 - 33033047/1162496a^8 - 13382337/581248a^7 + 44114819/1307808a^6 + 35533505/1307808a^5 - 22161593/653904a^4 - 8951479/326952a^3 + 423431/18164a^2 + 1252561/81738a - 342866/40869, 687083/669597696a^31 - 40499/41849856a^30 - 1290263/223199232a^29 + 109601/20924928a^28 + 13282951/669597696a^27 - 81417/4649984a^26 - 179791/2936832a^25 + 547397/10462464a^24 + 36321899/223199232a^23 - 2822359/20924928a^22 - 62269607/167399424a^21 + 1383697/4649984a^20 + 4256017/5873664a^19 - 23117263/41849856a^18 - 5578669/4649984a^17 + 8867449/10462464a^16 + 1101866515/669597696a^15 - 14110291/13949952a^14 - 2887453/1957888a^13 + 20655671/41849856a^12 - 3409519/13949952a^11 + 64243399/41849856a^10 + 706039/163476a^9 - 19791203/3487488a^8 - 1013173/91776a^7 + 31182233/2615616a^6 + 523175/27246a^5 - 3137663/163476a^4 - 9914695/326952a^3 + 786757/27246a^2 + 1118197/40869a - 1033307/40869, 419777/446398464a^31 + 178859/167399424a^30 - 340313/70483968a^29 - 1485277/167399424a^28 + 7499543/446398464a^27 + 6428147/167399424a^26 - 34911299/669597696a^25 - 2848447/20924928a^24 + 185677309/1339195392a^23 + 69609533/167399424a^22 - 71398213/223199232a^21 - 46607947/41849856a^20 + 428614339/669597696a^19 + 220907735/83699712a^18 - 371084111/334798848a^17 - 117007481/20924928a^16 + 738852529/446398464a^15 + 1790753435/167399424a^14 - 639955783/334798848a^13 - 1533241363/83699712a^12 + 103669951/83699712a^11 + 1164296249/41849856a^10 + 3260797/3487488a^9 - 388108465/10462464a^8 - 12519719/2615616a^7 + 27802867/653904a^6 + 6592223/653904a^5 - 26673571/653904a^4 - 1989065/108984a^3 + 982745/40869a^2 + 611641/40869*a + 106334/40869 (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:	$ 43671525522.024 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 43671525522.024 \cdot 16}{60\cdot\sqrt{87993561227221187133696000000000000000000000000}}\cr\approx \mathstrut & 0.23164319240278 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^32 - 5*x^30 + 19*x^28 - 70*x^26 + 215*x^24 - 590*x^22 + 1466*x^20 - 3340*x^18 + 7009*x^16 - 13360*x^14 + 23456*x^12 - 37760*x^10 + 55040*x^8 - 71680*x^6 + 77824*x^4 - 81920*x^2 + 65536)

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 - 5*x^30 + 19*x^28 - 70*x^26 + 215*x^24 - 590*x^22 + 1466*x^20 - 3340*x^18 + 7009*x^16 - 13360*x^14 + 23456*x^12 - 37760*x^10 + 55040*x^8 - 71680*x^6 + 77824*x^4 - 81920*x^2 + 65536, 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 - 5*x^30 + 19*x^28 - 70*x^26 + 215*x^24 - 590*x^22 + 1466*x^20 - 3340*x^18 + 7009*x^16 - 13360*x^14 + 23456*x^12 - 37760*x^10 + 55040*x^8 - 71680*x^6 + 77824*x^4 - 81920*x^2 + 65536);

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 - 5*x^30 + 19*x^28 - 70*x^26 + 215*x^24 - 590*x^22 + 1466*x^20 - 3340*x^18 + 7009*x^16 - 13360*x^14 + 23456*x^12 - 37760*x^10 + 55040*x^8 - 71680*x^6 + 77824*x^4 - 81920*x^2 + 65536);

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 $C_2^4:C_4$ is not computed

Intermediate fields

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	R	R	${\href{/padicField/7.4.0.1}{4} }^{8}$	${\href{/padicField/11.2.0.1}{2} }^{16}$	${\href{/padicField/13.4.0.1}{4} }^{8}$	${\href{/padicField/17.4.0.1}{4} }^{8}$	${\href{/padicField/19.2.0.1}{2} }^{16}$	${\href{/padicField/23.4.0.1}{4} }^{8}$	${\href{/padicField/29.2.0.1}{2} }^{16}$	${\href{/padicField/31.2.0.1}{2} }^{16}$	${\href{/padicField/37.4.0.1}{4} }^{8}$	R	${\href{/padicField/43.4.0.1}{4} }^{8}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.4.0.1}{4} }^{8}$	${\href{/padicField/59.2.0.1}{2} }^{16}$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
$3$ 3	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	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$5$ 5	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	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$41$ 41	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.4.2.1	$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	41.4.2.1	$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	41.4.2.1	$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	41.4.2.1	$x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more