Properties

Label 32.0.503...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $5.038\times 10^{48}$
Root discriminant $33.26$
Ramified primes $2, 3, 5, 17$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2\times C_4\times D_4$ (as 32T204)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536)
 
gp: K = bnfinit(x^32 - 2*x^31 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, -65536, 49152, -16384, 0, -24576, 17408, -12288, 768, 1024, 6464, -4096, 3328, -272, 156, -1452, 877, -726, 39, -34, 208, -128, 101, 8, 3, -24, 17, -12, 0, -2, 3, -2, 1]);
 

\( x^{32} - 2 x^{31} + 3 x^{30} - 2 x^{29} - 12 x^{27} + 17 x^{26} - 24 x^{25} + 3 x^{24} + 8 x^{23} + 101 x^{22} - 128 x^{21} + 208 x^{20} - 34 x^{19} + 39 x^{18} - 726 x^{17} + 877 x^{16} - 1452 x^{15} + 156 x^{14} - 272 x^{13} + 3328 x^{12} - 4096 x^{11} + 6464 x^{10} + 1024 x^{9} + 768 x^{8} - 12288 x^{7} + 17408 x^{6} - 24576 x^{5} - 16384 x^{3} + 49152 x^{2} - 65536 x + 65536 \)

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(5037920877776643425304576000000000000000000000000\)\(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 17^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $33.26$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 5, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $16$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{19} - \frac{1}{8} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} + \frac{1}{8} a^{13} + \frac{1}{4} a^{12} + \frac{3}{8} a^{11} - \frac{1}{4} a^{10} - \frac{3}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{20} + \frac{1}{16} a^{18} - \frac{1}{4} a^{17} - \frac{1}{2} a^{16} - \frac{1}{4} a^{15} + \frac{3}{16} a^{14} - \frac{1}{8} a^{13} - \frac{3}{16} a^{12} + \frac{1}{8} a^{11} + \frac{17}{48} a^{10} + \frac{3}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{16} a^{6} - \frac{1}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{96} a^{21} + \frac{1}{32} a^{19} - \frac{1}{8} a^{18} - \frac{1}{4} a^{17} - \frac{1}{8} a^{16} - \frac{13}{32} a^{15} - \frac{1}{16} a^{14} - \frac{3}{32} a^{13} + \frac{1}{16} a^{12} + \frac{17}{96} a^{11} + \frac{3}{16} a^{10} - \frac{1}{8} a^{9} - \frac{7}{16} a^{8} - \frac{15}{32} a^{7} - \frac{1}{2} a^{6} + \frac{15}{32} a^{5} + \frac{5}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{192} a^{22} - \frac{1}{192} a^{20} - \frac{1}{16} a^{19} + \frac{1}{16} a^{18} + \frac{3}{16} a^{17} - \frac{29}{64} a^{16} + \frac{7}{32} a^{15} - \frac{15}{64} a^{14} - \frac{11}{32} a^{13} - \frac{91}{192} a^{12} + \frac{15}{32} a^{11} - \frac{1}{6} a^{10} - \frac{3}{32} a^{9} - \frac{15}{64} a^{8} - \frac{3}{8} a^{7} + \frac{11}{64} a^{6} + \frac{5}{32} a^{5} - \frac{1}{16} a^{4} - \frac{3}{8} a^{3} + \frac{1}{12} a^{2} - \frac{1}{3}$, $\frac{1}{384} a^{23} - \frac{1}{384} a^{21} - \frac{1}{96} a^{20} + \frac{1}{32} a^{19} - \frac{3}{32} a^{18} + \frac{3}{128} a^{17} + \frac{23}{64} a^{16} + \frac{17}{128} a^{15} + \frac{1}{64} a^{14} + \frac{53}{384} a^{13} - \frac{13}{64} a^{12} + \frac{1}{24} a^{11} - \frac{85}{192} a^{10} + \frac{33}{128} a^{9} + \frac{5}{16} a^{8} + \frac{27}{128} a^{7} - \frac{23}{64} a^{6} - \frac{1}{32} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{768} a^{24} - \frac{1}{768} a^{22} - \frac{1}{192} a^{21} - \frac{1}{192} a^{20} - \frac{3}{64} a^{19} - \frac{13}{256} a^{18} - \frac{9}{128} a^{17} - \frac{111}{256} a^{16} - \frac{31}{128} a^{15} - \frac{91}{768} a^{14} - \frac{61}{128} a^{13} - \frac{7}{24} a^{12} - \frac{133}{384} a^{11} + \frac{211}{768} a^{10} + \frac{9}{32} a^{9} - \frac{37}{256} a^{8} + \frac{25}{128} a^{7} - \frac{5}{64} a^{6} + \frac{1}{4} a^{5} + \frac{7}{48} a^{4} - \frac{1}{4} a^{3} + \frac{5}{12} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{1536} a^{25} - \frac{1}{1536} a^{23} - \frac{1}{384} a^{22} - \frac{1}{384} a^{21} - \frac{1}{384} a^{20} - \frac{13}{512} a^{19} + \frac{7}{256} a^{18} + \frac{17}{512} a^{17} + \frac{97}{256} a^{16} - \frac{475}{1536} a^{15} - \frac{13}{256} a^{14} + \frac{11}{48} a^{13} - \frac{277}{768} a^{12} + \frac{403}{1536} a^{11} + \frac{95}{192} a^{10} + \frac{155}{512} a^{9} + \frac{89}{256} a^{8} - \frac{53}{128} a^{7} - \frac{5}{16} a^{6} - \frac{41}{96} a^{5} + \frac{5}{16} a^{4} + \frac{1}{3} a^{3} - \frac{5}{12} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{1373184} a^{26} + \frac{1}{171648} a^{25} + \frac{823}{1373184} a^{24} + \frac{13}{114432} a^{23} + \frac{169}{343296} a^{22} - \frac{1523}{343296} a^{21} + \frac{8953}{1373184} a^{20} - \frac{9469}{228864} a^{19} - \frac{2807}{457728} a^{18} + \frac{10345}{228864} a^{17} + \frac{646541}{1373184} a^{16} + \frac{220865}{686592} a^{15} + \frac{14959}{171648} a^{14} - \frac{39011}{228864} a^{13} + \frac{214163}{1373184} a^{12} - \frac{4669}{21456} a^{11} + \frac{556457}{1373184} a^{10} + \frac{80441}{228864} a^{9} + \frac{937}{114432} a^{8} + \frac{11353}{57216} a^{7} + \frac{413}{2682} a^{6} + \frac{16675}{42912} a^{5} - \frac{127}{10728} a^{4} - \frac{925}{3576} a^{3} - \frac{433}{2682} a^{2} - \frac{85}{2682} a + \frac{157}{1341}$, $\frac{1}{2746368} a^{27} + \frac{253}{915456} a^{25} + \frac{181}{686592} a^{24} - \frac{143}{686592} a^{23} - \frac{1087}{686592} a^{22} + \frac{473}{2746368} a^{21} + \frac{1}{9216} a^{20} - \frac{22951}{915456} a^{19} + \frac{10767}{152576} a^{18} + \frac{278717}{2746368} a^{17} + \frac{50685}{152576} a^{16} - \frac{13447}{38144} a^{15} + \frac{301855}{1373184} a^{14} - \frac{874237}{2746368} a^{13} + \frac{115025}{343296} a^{12} - \frac{1244087}{2746368} a^{11} + \frac{85999}{1373184} a^{10} + \frac{72533}{228864} a^{9} - \frac{743}{38144} a^{8} - \frac{30667}{85824} a^{7} - \frac{521}{2384} a^{6} - \frac{1333}{3576} a^{5} - \frac{1103}{10728} a^{4} - \frac{397}{1341} a^{3} + \frac{1591}{5364} a^{2} + \frac{472}{1341} a - \frac{628}{1341}$, $\frac{1}{38449152} a^{28} + \frac{1}{19224576} a^{27} - \frac{5}{38449152} a^{26} - \frac{1837}{6408192} a^{25} + \frac{1973}{4806144} a^{24} - \frac{1643}{1373184} a^{23} - \frac{11609}{5492736} a^{22} - \frac{16127}{3204096} a^{21} - \frac{90431}{12816384} a^{20} + \frac{111989}{3204096} a^{19} - \frac{677677}{5492736} a^{18} - \frac{1827365}{9612288} a^{17} + \frac{978989}{2403072} a^{16} + \frac{1469}{43008} a^{15} - \frac{3040801}{38449152} a^{14} - \frac{1209101}{19224576} a^{13} + \frac{7150469}{38449152} a^{12} - \frac{44875}{400512} a^{11} + \frac{143827}{457728} a^{10} - \frac{583157}{1602048} a^{9} - \frac{388285}{1201536} a^{8} + \frac{42985}{300384} a^{7} + \frac{20471}{42912} a^{6} + \frac{2737}{7152} a^{5} + \frac{4721}{18774} a^{4} + \frac{937}{37548} a^{3} + \frac{3287}{18774} a^{2} - \frac{843}{2086} a + \frac{373}{1043}$, $\frac{1}{2383847424} a^{29} + \frac{13}{1191923712} a^{28} - \frac{79}{794615808} a^{27} + \frac{25}{132435968} a^{26} + \frac{18209}{74495232} a^{25} - \frac{37327}{66217984} a^{24} + \frac{48191}{37838848} a^{23} + \frac{1527193}{595961856} a^{22} + \frac{1234429}{340549632} a^{21} + \frac{1453511}{595961856} a^{20} + \frac{110473373}{2383847424} a^{19} + \frac{32293267}{595961856} a^{18} + \frac{6813511}{33108992} a^{17} + \frac{58222167}{132435968} a^{16} - \frac{1074248401}{2383847424} a^{15} + \frac{26402679}{132435968} a^{14} + \frac{338108095}{794615808} a^{13} + \frac{13246097}{148990464} a^{12} + \frac{192172733}{595961856} a^{11} - \frac{10804051}{74495232} a^{10} - \frac{34551901}{74495232} a^{9} + \frac{683359}{2660544} a^{8} + \frac{1796261}{6207936} a^{7} - \frac{5493}{73904} a^{6} + \frac{202639}{9311904} a^{5} - \frac{10553}{64666} a^{4} + \frac{31847}{96999} a^{3} + \frac{132326}{290997} a^{2} + \frac{214321}{581994} a - \frac{98162}{290997}$, $\frac{1}{4767694848} a^{30} - \frac{5}{529743872} a^{28} - \frac{83}{1191923712} a^{27} + \frac{89}{595961856} a^{26} + \frac{17813}{1191923712} a^{25} + \frac{1764617}{4767694848} a^{24} + \frac{1286669}{2383847424} a^{23} + \frac{619407}{529743872} a^{22} - \frac{1662463}{794615808} a^{21} - \frac{31592527}{4767694848} a^{20} + \frac{4017521}{264871936} a^{19} - \frac{14768321}{397307904} a^{18} - \frac{235972133}{2383847424} a^{17} - \frac{2139106717}{4767694848} a^{16} - \frac{57621611}{297980928} a^{15} + \frac{1427736349}{4767694848} a^{14} - \frac{650459153}{2383847424} a^{13} + \frac{89377301}{198653952} a^{12} + \frac{34284473}{99326976} a^{11} + \frac{138967555}{297980928} a^{10} - \frac{2848789}{16554496} a^{9} + \frac{698255}{12415872} a^{8} + \frac{2019749}{9311904} a^{7} + \frac{3603437}{9311904} a^{6} + \frac{534445}{9311904} a^{5} - \frac{1309339}{4655952} a^{4} - \frac{712855}{2327976} a^{3} - \frac{463}{193998} a^{2} + \frac{44092}{96999} a + \frac{46559}{96999}$, $\frac{1}{9535389696} a^{31} - \frac{1}{9535389696} a^{29} + \frac{17}{2383847424} a^{28} - \frac{197}{2383847424} a^{27} - \frac{611}{2383847424} a^{26} - \frac{724279}{9535389696} a^{25} - \frac{698153}{1589231616} a^{24} - \frac{978319}{3178463232} a^{23} - \frac{1402873}{4767694848} a^{22} + \frac{2443901}{1059487744} a^{21} - \frac{1527295}{227033088} a^{20} - \frac{12304021}{297980928} a^{19} - \frac{468423601}{4767694848} a^{18} - \frac{2141092141}{9535389696} a^{17} + \frac{18693379}{1191923712} a^{16} + \frac{283693441}{9535389696} a^{15} - \frac{254467377}{529743872} a^{14} + \frac{1418833}{113516544} a^{13} + \frac{343342645}{1191923712} a^{12} - \frac{32342927}{66217984} a^{11} + \frac{5905783}{33108992} a^{10} - \frac{10560919}{148990464} a^{9} - \frac{32321279}{74495232} a^{8} - \frac{3720133}{18623808} a^{7} + \frac{8143193}{18623808} a^{6} - \frac{2264785}{9311904} a^{5} - \frac{471277}{1551984} a^{4} - \frac{14351}{96999} a^{3} - \frac{363739}{1163988} a^{2} + \frac{195613}{581994} a - \frac{47933}{96999}$

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

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);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{47}{4571136} a^{31} + \frac{5633}{340549632} a^{30} - \frac{4229}{529743872} a^{29} + \frac{43541}{2383847424} a^{28} + \frac{941}{397307904} a^{27} - \frac{38225}{198653952} a^{26} - \frac{1284011}{4767694848} a^{25} - \frac{11399}{297980928} a^{24} - \frac{158635}{681099264} a^{23} - \frac{904063}{1191923712} a^{22} + \frac{6664165}{4767694848} a^{21} + \frac{1070059}{397307904} a^{20} + \frac{3985}{4272128} a^{19} + \frac{6172933}{2383847424} a^{18} + \frac{4386553}{1589231616} a^{17} - \frac{5523821}{794615808} a^{16} - \frac{11069533}{681099264} a^{15} - \frac{83371}{10642176} a^{14} - \frac{2893957}{148990464} a^{13} - \frac{25379927}{1191923712} a^{12} + \frac{8495279}{297980928} a^{11} + \frac{583663}{8277248} a^{10} + \frac{251537}{8277248} a^{9} + \frac{1776961}{18623808} a^{8} + \frac{906827}{6207936} a^{7} - \frac{4195}{73904} a^{6} - \frac{288347}{1163988} a^{5} - \frac{43541}{1163988} a^{4} - \frac{293827}{1163988} a^{3} - \frac{441451}{1163988} a^{2} - \frac{1051}{9387} a + \frac{163204}{290997} \) (order $30$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 190465720691.2432 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 190465720691.2432 \cdot 16}{30\sqrt{5037920877776643425304576000000000000000000000000}}\approx 0.267034746152232$ (assuming GRH)

Galois group

$C_2\times C_4\times D_4$ (as 32T204):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 64
The 40 conjugacy class representatives for $C_2\times C_4\times D_4$
Character table for $C_2\times C_4\times D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{15})^+\), 4.0.8000.2, \(\Q(\zeta_{5})\), 4.4.72000.1, \(\Q(\sqrt{2}, \sqrt{-15})\), 4.0.27200.2, 4.4.9792.1, \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.244800.1, 4.0.1088.2, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-6}, \sqrt{10})\), 8.0.59927040000.35, 8.0.207360000.1, 8.0.59927040000.32, 8.0.59927040000.12, 8.0.739840000.6, 8.0.95883264.1, 8.8.59927040000.2, 8.0.5184000000.1, 8.0.5184000000.5, 8.0.1498176000000.24, 8.8.18496000000.1, 8.8.5184000000.1, 8.0.64000000.2, \(\Q(\zeta_{15})\), 8.0.5184000000.3, 16.0.3591250123161600000000.1, Deg 16, 16.0.26873856000000000000.2, Deg 16, 16.0.342102016000000000000.1, Deg 16, Deg 16

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 32 siblings: data not computed

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$