↑

Properties

Label 32.0.15729269092...000.10
Degree $32$
Signature $[0, 16]$
Discriminant $2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $61.29$
Ramified primes $2, 3, 5, 7$
Class number $256$ (GRH)
Class group $[2, 4, 4, 8]$ (GRH)
Galois group $C_2^3\times C_4$ (as 32T34)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14281, -285676, 2765566, -17245236, 77802183, -270487448, 753847668, -1730268776, 3335315605, -5480415412, 7766700502, -9583863764, 10378954300, -9930099548, 8440471994, -6403659660, 4353318135, -2660057452, 1464404298, -727469324, 326361852, -132207604, 48314366, -15889444, 4688282, -1233388, 287602, -58604, 10334, -1520, 184, -16, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 184*x^30 - 1520*x^29 + 10334*x^28 - 58604*x^27 + 287602*x^26 - 1233388*x^25 + 4688282*x^24 - 15889444*x^23 + 48314366*x^22 - 132207604*x^21 + 326361852*x^20 - 727469324*x^19 + 1464404298*x^18 - 2660057452*x^17 + 4353318135*x^16 - 6403659660*x^15 + 8440471994*x^14 - 9930099548*x^13 + 10378954300*x^12 - 9583863764*x^11 + 7766700502*x^10 - 5480415412*x^9 + 3335315605*x^8 - 1730268776*x^7 + 753847668*x^6 - 270487448*x^5 + 77802183*x^4 - 17245236*x^3 + 2765566*x^2 - 285676*x + 14281)
 
gp: K = bnfinit(x^32 - 16*x^31 + 184*x^30 - 1520*x^29 + 10334*x^28 - 58604*x^27 + 287602*x^26 - 1233388*x^25 + 4688282*x^24 - 15889444*x^23 + 48314366*x^22 - 132207604*x^21 + 326361852*x^20 - 727469324*x^19 + 1464404298*x^18 - 2660057452*x^17 + 4353318135*x^16 - 6403659660*x^15 + 8440471994*x^14 - 9930099548*x^13 + 10378954300*x^12 - 9583863764*x^11 + 7766700502*x^10 - 5480415412*x^9 + 3335315605*x^8 - 1730268776*x^7 + 753847668*x^6 - 270487448*x^5 + 77802183*x^4 - 17245236*x^3 + 2765566*x^2 - 285676*x + 14281, 1)
 

Normalized defining polynomial

\( x^{32} - 16 x^{31} + 184 x^{30} - 1520 x^{29} + 10334 x^{28} - 58604 x^{27} + 287602 x^{26} - 1233388 x^{25} + 4688282 x^{24} - 15889444 x^{23} + 48314366 x^{22} - 132207604 x^{21} + 326361852 x^{20} - 727469324 x^{19} + 1464404298 x^{18} - 2660057452 x^{17} + 4353318135 x^{16} - 6403659660 x^{15} + 8440471994 x^{14} - 9930099548 x^{13} + 10378954300 x^{12} - 9583863764 x^{11} + 7766700502 x^{10} - 5480415412 x^{9} + 3335315605 x^{8} - 1730268776 x^{7} + 753847668 x^{6} - 270487448 x^{5} + 77802183 x^{4} - 17245236 x^{3} + 2765566 x^{2} - 285676 x + 14281 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.29$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(517,·)$, $\chi_{840}(769,·)$, $\chi_{840}(139,·)$, $\chi_{840}(13,·)$, $\chi_{840}(29,·)$, $\chi_{840}(671,·)$, $\chi_{840}(169,·)$, $\chi_{840}(811,·)$, $\chi_{840}(827,·)$, $\chi_{840}(701,·)$, $\chi_{840}(629,·)$, $\chi_{840}(323,·)$, $\chi_{840}(71,·)$, $\chi_{840}(713,·)$, $\chi_{840}(587,·)$, $\chi_{840}(461,·)$, $\chi_{840}(463,·)$, $\chi_{840}(211,·)$, $\chi_{840}(727,·)$, $\chi_{840}(601,·)$, $\chi_{840}(223,·)$, $\chi_{840}(617,·)$, $\chi_{840}(239,·)$, $\chi_{840}(113,·)$, $\chi_{840}(839,·)$, $\chi_{840}(83,·)$, $\chi_{840}(757,·)$, $\chi_{840}(377,·)$, $\chi_{840}(379,·)$, $\chi_{840}(253,·)$, $\chi_{840}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{5} a^{24} - \frac{2}{5} a^{23} - \frac{2}{5} a^{22} - \frac{2}{5} a^{21} + \frac{1}{5} a^{20} - \frac{2}{5} a^{19} + \frac{1}{5} a^{18} + \frac{1}{5} a^{17} - \frac{1}{5} a^{16} + \frac{1}{5} a^{15} + \frac{1}{5} a^{14} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{25} - \frac{1}{5} a^{23} - \frac{1}{5} a^{22} + \frac{2}{5} a^{21} + \frac{2}{5} a^{19} - \frac{2}{5} a^{18} + \frac{1}{5} a^{17} - \frac{1}{5} a^{16} - \frac{2}{5} a^{15} + \frac{2}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{55} a^{26} - \frac{2}{55} a^{25} - \frac{1}{11} a^{24} - \frac{16}{55} a^{23} - \frac{13}{55} a^{22} - \frac{1}{5} a^{21} + \frac{13}{55} a^{20} + \frac{2}{55} a^{19} - \frac{9}{55} a^{18} + \frac{13}{55} a^{17} - \frac{1}{5} a^{16} + \frac{17}{55} a^{15} + \frac{13}{55} a^{14} + \frac{27}{55} a^{13} + \frac{8}{55} a^{12} + \frac{4}{55} a^{11} + \frac{5}{11} a^{10} + \frac{17}{55} a^{9} + \frac{2}{11} a^{8} - \frac{2}{11} a^{7} - \frac{3}{11} a^{6} + \frac{27}{55} a^{5} + \frac{9}{55} a^{4} - \frac{23}{55} a^{3} - \frac{9}{55} a^{2} + \frac{4}{55} a - \frac{18}{55}$, $\frac{1}{55} a^{27} + \frac{2}{55} a^{25} - \frac{4}{55} a^{24} + \frac{2}{11} a^{23} + \frac{18}{55} a^{22} + \frac{24}{55} a^{21} - \frac{1}{11} a^{20} - \frac{27}{55} a^{19} - \frac{1}{11} a^{18} - \frac{7}{55} a^{17} + \frac{17}{55} a^{16} - \frac{8}{55} a^{15} - \frac{13}{55} a^{14} + \frac{18}{55} a^{13} - \frac{24}{55} a^{12} + \frac{2}{5} a^{11} + \frac{1}{55} a^{10} + \frac{2}{5} a^{9} - \frac{1}{55} a^{8} - \frac{13}{55} a^{7} + \frac{19}{55} a^{6} + \frac{19}{55} a^{5} - \frac{27}{55} a^{4} + \frac{8}{55} a^{2} - \frac{21}{55} a + \frac{8}{55}$, $\frac{1}{3355} a^{28} - \frac{14}{3355} a^{27} + \frac{2}{3355} a^{26} + \frac{2}{55} a^{25} + \frac{4}{61} a^{24} + \frac{516}{3355} a^{23} + \frac{49}{671} a^{22} - \frac{146}{305} a^{21} - \frac{1618}{3355} a^{20} - \frac{1442}{3355} a^{19} - \frac{641}{3355} a^{18} - \frac{842}{3355} a^{17} - \frac{1489}{3355} a^{16} + \frac{26}{61} a^{15} + \frac{717}{3355} a^{14} - \frac{1662}{3355} a^{13} - \frac{357}{3355} a^{12} + \frac{1651}{3355} a^{11} + \frac{204}{671} a^{10} + \frac{813}{3355} a^{9} + \frac{130}{671} a^{8} + \frac{421}{3355} a^{7} - \frac{111}{671} a^{6} - \frac{678}{3355} a^{5} - \frac{348}{3355} a^{4} + \frac{558}{3355} a^{3} - \frac{273}{671} a^{2} + \frac{45}{671} a + \frac{26}{671}$, $\frac{1}{3355} a^{29} - \frac{1}{305} a^{27} + \frac{28}{3355} a^{26} - \frac{146}{3355} a^{25} + \frac{119}{3355} a^{24} + \frac{103}{671} a^{23} - \frac{677}{3355} a^{22} - \frac{251}{3355} a^{21} + \frac{49}{671} a^{20} - \frac{1187}{3355} a^{19} - \frac{182}{671} a^{18} + \frac{1302}{3355} a^{17} + \frac{1141}{3355} a^{16} - \frac{918}{3355} a^{15} - \frac{87}{305} a^{14} + \frac{531}{3355} a^{13} + \frac{679}{3355} a^{12} + \frac{832}{3355} a^{11} + \frac{148}{3355} a^{10} - \frac{778}{3355} a^{9} + \frac{6}{305} a^{8} - \frac{47}{305} a^{7} - \frac{457}{3355} a^{6} - \frac{263}{3355} a^{5} - \frac{959}{3355} a^{4} - \frac{812}{3355} a^{3} - \frac{1561}{3355} a^{2} - \frac{1051}{3355} a + \frac{112}{3355}$, $\frac{1}{15438568712622319807244405} a^{30} - \frac{3}{3087713742524463961448881} a^{29} + \frac{406360536594718915140}{3087713742524463961448881} a^{28} - \frac{93263073972558439537}{50618258074171540351621} a^{27} - \frac{31557578805981535471652}{15438568712622319807244405} a^{26} - \frac{1013418820691330042890358}{15438568712622319807244405} a^{25} + \frac{283784318873338264865659}{15438568712622319807244405} a^{24} + \frac{2835641551812527738870488}{15438568712622319807244405} a^{23} - \frac{1832693557254521827898838}{15438568712622319807244405} a^{22} - \frac{326436564272268504953809}{1403506246602029073385855} a^{21} + \frac{4150676148953724861250729}{15438568712622319807244405} a^{20} + \frac{26249799356932267715351}{3087713742524463961448881} a^{19} + \frac{1455950272436973708938392}{3087713742524463961448881} a^{18} - \frac{493545881229691822873551}{1403506246602029073385855} a^{17} - \frac{5086642834455762839161073}{15438568712622319807244405} a^{16} - \frac{3414026859421917176436439}{15438568712622319807244405} a^{15} - \frac{7057940400822981735537}{50618258074171540351621} a^{14} + \frac{98813531813989165999597}{15438568712622319807244405} a^{13} + \frac{2511829424286478037718907}{15438568712622319807244405} a^{12} - \frac{799259088732333213821305}{3087713742524463961448881} a^{11} - \frac{618388894402140490979353}{15438568712622319807244405} a^{10} - \frac{2495312339454563535913246}{15438568712622319807244405} a^{9} + \frac{192979504938391279430914}{3087713742524463961448881} a^{8} + \frac{2062039036817579074312626}{15438568712622319807244405} a^{7} - \frac{985580767795970398357594}{3087713742524463961448881} a^{6} - \frac{5483874548212463818140749}{15438568712622319807244405} a^{5} + \frac{3859719442817885610064499}{15438568712622319807244405} a^{4} + \frac{5003002300223148427692856}{15438568712622319807244405} a^{3} + \frac{116121464275686559594214}{3087713742524463961448881} a^{2} + \frac{1449856433965674974418090}{3087713742524463961448881} a + \frac{4967326869216241833254271}{15438568712622319807244405}$, $\frac{1}{559509168714145492134344481605} a^{31} + \frac{3621}{111901833742829098426868896321} a^{30} + \frac{645672082958342118180943}{50864469883104135648576771055} a^{29} + \frac{17216960142515216013493932}{559509168714145492134344481605} a^{28} + \frac{2996914055612266183405406272}{559509168714145492134344481605} a^{27} + \frac{3943876437771499039640782342}{559509168714145492134344481605} a^{26} - \frac{5030140314686516276799585698}{559509168714145492134344481605} a^{25} - \frac{21775526173381369727686823901}{559509168714145492134344481605} a^{24} - \frac{72242305904905123651990961687}{559509168714145492134344481605} a^{23} + \frac{39473082784960910351576250285}{111901833742829098426868896321} a^{22} + \frac{148384145905153125141069572491}{559509168714145492134344481605} a^{21} - \frac{267721946687276210970317157541}{559509168714145492134344481605} a^{20} - \frac{186039168663991239857384913672}{559509168714145492134344481605} a^{19} - \frac{122811150100572563398890699304}{559509168714145492134344481605} a^{18} + \frac{24565904639685999380408740175}{111901833742829098426868896321} a^{17} - \frac{101393092949343597265500031381}{559509168714145492134344481605} a^{16} - \frac{37554569604919113415236599423}{559509168714145492134344481605} a^{15} + \frac{61550536386545227786431540697}{559509168714145492134344481605} a^{14} - \frac{9695104353587905946092079129}{559509168714145492134344481605} a^{13} - \frac{179560156164492467315462269138}{559509168714145492134344481605} a^{12} - \frac{52185704237692680763201105504}{559509168714145492134344481605} a^{11} - \frac{53300557028636314304553590126}{559509168714145492134344481605} a^{10} + \frac{80701187519259718598646441204}{559509168714145492134344481605} a^{9} - \frac{74068742935061747910615870771}{559509168714145492134344481605} a^{8} + \frac{96719328915093542064552212537}{559509168714145492134344481605} a^{7} - \frac{16885198860105192559268271954}{111901833742829098426868896321} a^{6} - \frac{118699928408787908076629095919}{559509168714145492134344481605} a^{5} - \frac{15186247797545351181140720907}{50864469883104135648576771055} a^{4} + \frac{14616340071148150819739132904}{50864469883104135648576771055} a^{3} + \frac{234161968673296052566409687186}{559509168714145492134344481605} a^{2} + \frac{191907415871919532673267280659}{559509168714145492134344481605} a + \frac{191339507895585902401791329181}{559509168714145492134344481605}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{8}$, which has order $256$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9542212134222.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{-2}, \sqrt{-105})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{70})\), \(\Q(\sqrt{-6}, \sqrt{70})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-10}, \sqrt{14})\), \(\Q(\sqrt{-30}, \sqrt{-35})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{15}, \sqrt{42})\), \(\Q(\sqrt{-21}, \sqrt{-30})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{-7}, \sqrt{-30})\), \(\Q(\sqrt{-10}, \sqrt{-21})\), \(\Q(\sqrt{14}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-7}, \sqrt{15})\), \(\Q(\sqrt{-10}, \sqrt{42})\), \(\Q(\sqrt{14}, \sqrt{-30})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\sqrt{3}, \sqrt{14})\), 4.0.8000.2, 4.4.392000.1, \(\Q(\zeta_{20})^+\), 4.0.98000.1, 4.0.55125.1, \(\Q(\zeta_{15})^+\), 4.4.3528000.1, 4.0.72000.2, 8.0.7965941760000.35, 8.0.6146560000.1, 8.0.7965941760000.17, 8.0.497871360000.9, 8.0.7965941760000.51, 8.0.31116960000.6, 8.0.7965941760000.6, 8.0.7965941760000.55, 8.0.7965941760000.26, 8.0.3317760000.7, 8.0.12745506816.2, 8.8.7965941760000.2, 8.0.7965941760000.19, 8.0.7965941760000.18, 8.0.7965941760000.11, 8.0.153664000000.2, 8.0.9604000000.2, 8.0.3038765625.2, 8.0.12446784000000.10, 8.0.1024000000.1, 8.0.2458624000000.5, 8.0.12446784000000.7, 8.0.5184000000.2, 8.0.2458624000000.1, 8.8.2458624000000.2, 8.0.12446784000000.4, 8.8.12446784000000.5, 8.0.12446784000000.11, 8.8.12446784000000.6, 8.8.199148544000000.7, 8.0.199148544000000.140, 8.0.5184000000.1, 8.0.12446784000000.8, 8.0.82944000000.2, 8.0.199148544000000.67, 8.0.199148544000000.138, 8.0.199148544000000.118, 8.0.777924000000.6, 8.0.777924000000.7, 8.0.82944000000.1, 8.8.199148544000000.5, \(\Q(\zeta_{60})^+\), 8.0.777924000000.5, 16.0.63456228123711897600000000.24, 16.0.6044831973376000000000000.9, 16.0.154922431942656000000000000.12, 16.0.154922431942656000000000000.18, 16.0.39660142577319936000000000000.70, 16.0.39660142577319936000000000000.67, 16.0.605165749776000000000000.2, 16.0.39660142577319936000000000000.12, 16.0.39660142577319936000000000000.28, 16.0.6879707136000000000000.1, 16.0.39660142577319936000000000000.8, 16.0.39660142577319936000000000000.3, 16.16.39660142577319936000000000000.1, 16.0.39660142577319936000000000000.26, 16.0.39660142577319936000000000000.9

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

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 R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$