sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 7*x^28 - 704*x^24 - 5047*x^20 + 565969*x^16 - 80752*x^12 - 180224*x^8 - 28672*x^4 + 65536)
gp: K = bnfinit(y^32 - 7*y^28 - 704*y^24 - 5047*y^20 + 565969*y^16 - 80752*y^12 - 180224*y^8 - 28672*y^4 + 65536, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 7*x^28 - 704*x^24 - 5047*x^20 + 565969*x^16 - 80752*x^12 - 180224*x^8 - 28672*x^4 + 65536);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 7*x^28 - 704*x^24 - 5047*x^20 + 565969*x^16 - 80752*x^12 - 180224*x^8 - 28672*x^4 + 65536)
\( x^{32} - 7x^{28} - 704x^{24} - 5047x^{20} + 565969x^{16} - 80752x^{12} - 180224x^{8} - 28672x^{4} + 65536 \)
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sage: K.defining_polynomial()
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
| Degree: | | $32$ |
|
| Signature: | | $[0, 16]$ |
|
| Discriminant: | |
\(4026692887688564776141139207792885760000000000000000\)
\(\medspace = 2^{64}\cdot 3^{16}\cdot 5^{16}\cdot 7^{16}\)
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|
magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK) |
| Root discriminant: | | \(40.99\) | 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: | | $2^{2}3^{1/2}5^{1/2}7^{1/2}\approx 40.98780306383839$
|
| Ramified primes: | |
\(2\), \(3\), \(5\), \(7\)
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|
gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK))) |
| Discriminant root field: | | \(\Q\)
|
| $\card{ \Gal(K/\Q) }$: | | $32$ |
|
| This field is Galois and abelian over $\Q$. |
| Conductor: | | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) |
| Dirichlet character group:
| |
$\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(391,·)$, $\chi_{840}(139,·)$, $\chi_{840}(659,·)$, $\chi_{840}(281,·)$, $\chi_{840}(29,·)$, $\chi_{840}(671,·)$, $\chi_{840}(419,·)$, $\chi_{840}(421,·)$, $\chi_{840}(41,·)$, $\chi_{840}(811,·)$, $\chi_{840}(559,·)$, $\chi_{840}(181,·)$, $\chi_{840}(799,·)$, $\chi_{840}(769,·)$, $\chi_{840}(701,·)$, $\chi_{840}(449,·)$, $\chi_{840}(839,·)$, $\chi_{840}(631,·)$, $\chi_{840}(589,·)$, $\chi_{840}(461,·)$, $\chi_{840}(209,·)$, $\chi_{840}(211,·)$, $\chi_{840}(601,·)$, $\chi_{840}(71,·)$, $\chi_{840}(349,·)$, $\chi_{840}(379,·)$, $\chi_{840}(491,·)$, $\chi_{840}(239,·)$, $\chi_{840}(629,·)$, $\chi_{840}(169,·)$, $\chi_{840}(251,·)$$\rbrace$
|
| This is a CM field. |
| Reflex fields: | | unavailable$^{32768}$ |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{12}+\frac{1}{9}$, $\frac{1}{18}a^{13}+\frac{1}{18}a$, $\frac{1}{36}a^{14}+\frac{1}{36}a^{2}$, $\frac{1}{72}a^{15}+\frac{1}{72}a^{3}$, $\frac{1}{432}a^{16}+\frac{1}{27}a^{12}+\frac{1}{9}a^{8}+\frac{169}{432}a^{4}+\frac{4}{27}$, $\frac{1}{432}a^{17}-\frac{1}{54}a^{13}+\frac{1}{9}a^{9}+\frac{169}{432}a^{5}+\frac{5}{54}a$, $\frac{1}{432}a^{18}+\frac{1}{108}a^{14}+\frac{1}{9}a^{10}+\frac{169}{432}a^{6}+\frac{13}{108}a^{2}$, $\frac{1}{432}a^{19}-\frac{1}{216}a^{15}+\frac{1}{9}a^{11}+\frac{169}{432}a^{7}+\frac{23}{216}a^{3}$, $\frac{1}{432}a^{20}-\frac{1}{27}a^{12}-\frac{23}{432}a^{8}-\frac{4}{9}a^{4}+\frac{11}{27}$, $\frac{1}{864}a^{21}-\frac{1}{864}a^{17}+\frac{1}{54}a^{13}+\frac{73}{864}a^{9}+\frac{359}{864}a^{5}-\frac{4}{27}a$, $\frac{1}{1728}a^{22}+\frac{1}{1728}a^{18}-\frac{119}{1728}a^{10}-\frac{743}{1728}a^{6}+\frac{11}{36}a^{2}$, $\frac{1}{3456}a^{23}+\frac{1}{3456}a^{19}+\frac{457}{3456}a^{11}+\frac{409}{3456}a^{7}+\frac{23}{72}a^{3}$, $\frac{1}{9020160}a^{24}+\frac{7}{601344}a^{20}-\frac{41}{37584}a^{16}-\frac{419767}{9020160}a^{12}-\frac{79313}{601344}a^{8}-\frac{4555}{18792}a^{4}+\frac{6541}{35235}$, $\frac{1}{18040320}a^{25}+\frac{7}{1202688}a^{21}-\frac{41}{75168}a^{17}-\frac{419767}{18040320}a^{13}-\frac{79313}{1202688}a^{9}+\frac{14237}{37584}a^{5}+\frac{6541}{70470}a$, $\frac{1}{36080640}a^{26}+\frac{7}{2405376}a^{22}+\frac{133}{150336}a^{18}+\frac{248393}{36080640}a^{14}+\frac{54319}{2405376}a^{10}+\frac{7235}{18792}a^{6}+\frac{16981}{140940}a^{2}$, $\frac{1}{72161280}a^{27}+\frac{7}{4810752}a^{23}+\frac{133}{300672}a^{19}+\frac{248393}{72161280}a^{15}-\frac{747473}{4810752}a^{11}-\frac{5293}{37584}a^{7}-\frac{29999}{281880}a^{3}$, $\frac{1}{417484755763200}a^{28}-\frac{9168791}{417484755763200}a^{24}+\frac{1806059933}{1739519815680}a^{20}-\frac{399491321527}{417484755763200}a^{16}+\frac{4033053919937}{417484755763200}a^{12}-\frac{8020821793}{108719988480}a^{8}-\frac{18239175839}{101924989200}a^{4}+\frac{4977957769}{101924989200}$, $\frac{1}{834969511526400}a^{29}-\frac{9168791}{834969511526400}a^{25}+\frac{1806059933}{3479039631360}a^{21}-\frac{399491321527}{834969511526400}a^{17}+\frac{4033053919937}{834969511526400}a^{13}+\frac{28219174367}{217439976960}a^{9}+\frac{49710816961}{203849978400}a^{5}-\frac{62972035031}{203849978400}a$, $\frac{1}{16\!\cdots\!00}a^{30}-\frac{9168791}{16\!\cdots\!00}a^{26}+\frac{1806059933}{6958079262720}a^{22}+\frac{1533308473673}{16\!\cdots\!00}a^{18}-\frac{11429344441663}{16\!\cdots\!00}a^{14}+\frac{52379171807}{434879953920}a^{10}+\frac{129457683511}{407699956800}a^{6}-\frac{44097037031}{407699956800}a^{2}$, $\frac{1}{33\!\cdots\!00}a^{31}-\frac{9168791}{33\!\cdots\!00}a^{27}+\frac{1806059933}{13916158525440}a^{23}-\frac{2332291116727}{33\!\cdots\!00}a^{19}+\frac{19495452281537}{33\!\cdots\!00}a^{15}-\frac{140900807713}{869759907840}a^{11}-\frac{301836020789}{815399913600}a^{7}-\frac{217747018631}{815399913600}a^{3}$
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not computed
sage: K.class_group().invariants()
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | | $15$
|
|
| Torsion generator: | |
\( \frac{10002394073}{3339878046105600} a^{31} - \frac{47178166783}{3339878046105600} a^{27} - \frac{30022469771}{13916158525440} a^{23} - \frac{66534600394271}{3339878046105600} a^{19} + \frac{5548482660848281}{3339878046105600} a^{15} + \frac{3160937174461}{869759907840} a^{11} - \frac{708699417311}{407699956800} a^{7} - \frac{1076519987503}{815399913600} a^{3} \)
(order $24$)
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| sage: UK.torsion_generator() magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK) |
| Fundamental units: | | not computed
| sage: UK.fundamental_units() magma: [K|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)] |
| Regulator: | | not computed
|
|
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
not computed
\end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^32 - 7*x^28 - 704*x^24 - 5047*x^20 + 565969*x^16 - 80752*x^12 - 180224*x^8 - 28672*x^4 + 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 - 7*x^28 - 704*x^24 - 5047*x^20 + 565969*x^16 - 80752*x^12 - 180224*x^8 - 28672*x^4 + 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 - 7*x^28 - 704*x^24 - 5047*x^20 + 565969*x^16 - 80752*x^12 - 180224*x^8 - 28672*x^4 + 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 - 7*x^28 - 704*x^24 - 5047*x^20 + 565969*x^16 - 80752*x^12 - 180224*x^8 - 28672*x^4 + 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))))
$C_2^5$ (as 32T39):
sage: K.galois_group(type='pari') magma: G = GaloisGroup(K); oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
|
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{-210}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{210})\), \(\Q(i, \sqrt{105})\), \(\Q(\sqrt{-2}, \sqrt{-105})\), \(\Q(\sqrt{-2}, \sqrt{105})\), \(\Q(\sqrt{2}, \sqrt{105})\), \(\Q(\sqrt{2}, \sqrt{-105})\), \(\Q(i, \sqrt{70})\), \(\Q(i, \sqrt{35})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{35})\), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\sqrt{2}, \sqrt{35})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-3}, \sqrt{-70})\), \(\Q(\sqrt{3}, \sqrt{70})\), \(\Q(\sqrt{6}, \sqrt{35})\), \(\Q(\sqrt{-6}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{-70})\), \(\Q(\sqrt{-3}, \sqrt{70})\), \(\Q(\sqrt{-6}, \sqrt{35})\), \(\Q(\sqrt{6}, \sqrt{-35})\), \(\Q(\sqrt{6}, \sqrt{-70})\), \(\Q(\sqrt{-6}, \sqrt{70})\), \(\Q(\sqrt{-3}, \sqrt{35})\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-70})\), \(\Q(\sqrt{6}, \sqrt{70})\), \(\Q(\sqrt{3}, \sqrt{35})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(i, \sqrt{14})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{30})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{42})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{14}, \sqrt{15})\), \(\Q(\sqrt{-14}, \sqrt{-15})\), \(\Q(\sqrt{-7}, \sqrt{-30})\), \(\Q(\sqrt{7}, \sqrt{30})\), \(\Q(\sqrt{-5}, \sqrt{-42})\), \(\Q(\sqrt{5}, \sqrt{42})\), \(\Q(\sqrt{10}, \sqrt{21})\), \(\Q(\sqrt{-10}, \sqrt{-21})\), \(\Q(\sqrt{14}, \sqrt{-15})\), \(\Q(\sqrt{-14}, \sqrt{15})\), \(\Q(\sqrt{-7}, \sqrt{30})\), \(\Q(\sqrt{7}, \sqrt{-30})\), \(\Q(\sqrt{-5}, \sqrt{42})\), \(\Q(\sqrt{5}, \sqrt{-42})\), \(\Q(\sqrt{10}, \sqrt{-21})\), \(\Q(\sqrt{-10}, \sqrt{21})\), \(\Q(\sqrt{14}, \sqrt{-30})\), \(\Q(\sqrt{-14}, \sqrt{30})\), \(\Q(\sqrt{-7}, \sqrt{15})\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{10}, \sqrt{-42})\), \(\Q(\sqrt{-10}, \sqrt{42})\), \(\Q(\sqrt{14}, \sqrt{30})\), \(\Q(\sqrt{-14}, \sqrt{-30})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{7}, \sqrt{15})\), \(\Q(\sqrt{-5}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{10}, \sqrt{42})\), \(\Q(\sqrt{-10}, \sqrt{-42})\), \(\Q(\sqrt{-5}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{7}, \sqrt{-10})\), \(\Q(\sqrt{15}, \sqrt{-42})\), \(\Q(\sqrt{-15}, \sqrt{42})\), \(\Q(\sqrt{21}, \sqrt{-30})\), \(\Q(\sqrt{-21}, \sqrt{30})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{-5}, \sqrt{-14})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{7}, \sqrt{10})\), \(\Q(\sqrt{15}, \sqrt{42})\), \(\Q(\sqrt{-15}, \sqrt{-42})\), \(\Q(\sqrt{-21}, \sqrt{-30})\), \(\Q(\sqrt{21}, \sqrt{30})\), \(\Q(\sqrt{10}, \sqrt{14})\), \(\Q(\sqrt{-10}, \sqrt{-14})\), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{15}, \sqrt{21})\), \(\Q(\sqrt{-15}, \sqrt{-21})\), \(\Q(\sqrt{-30}, \sqrt{35})\), \(\Q(\sqrt{30}, \sqrt{35})\), \(\Q(\sqrt{-10}, \sqrt{14})\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{-30}, \sqrt{-35})\), \(\Q(\sqrt{30}, \sqrt{-35})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), 8.0.7965941760000.69, 8.0.98344960000.9, \(\Q(\zeta_{24})\), 8.0.7965941760000.70, 8.0.7965941760000.54, 8.0.7965941760000.44, 8.0.31116960000.10, 8.0.7965941760000.66, 8.0.7965941760000.35, 8.0.7965941760000.57, 8.0.497871360000.17, 8.0.497871360000.14, 8.8.7965941760000.5, 8.0.7965941760000.58, 8.0.7965941760000.67, 8.0.157351936.1, 8.0.3317760000.4, 8.0.40960000.1, 8.0.12745506816.8, 8.0.7965941760000.41, 8.0.7965941760000.65, 8.0.7965941760000.68, 8.0.7965941760000.39, 8.0.7965941760000.3, 8.0.31116960000.7, 8.0.31116960000.8, 8.0.7965941760000.13, 8.0.7965941760000.26, 8.0.7965941760000.10, 8.0.7965941760000.12, 8.0.7965941760000.55, 8.0.497871360000.3, 8.0.7965941760000.24, 8.0.7965941760000.4, 8.0.497871360000.13, 8.8.7965941760000.4, 8.0.497871360000.10, 8.0.7965941760000.33, 8.8.497871360000.1, 8.0.7965941760000.38, 8.0.7965941760000.47, 8.0.7965941760000.40, 8.0.7965941760000.62, 8.0.98344960000.8, 8.0.98344960000.7, 8.0.7965941760000.63, 8.0.7965941760000.21, 8.0.98344960000.2, 8.0.384160000.1, 8.0.31116960000.4, 8.0.7965941760000.50, 8.0.12745506816.9, 8.0.49787136.1, 8.0.12960000.1, 8.0.3317760000.2, 8.0.12745506816.1, 8.0.12745506816.7, 8.0.3317760000.5, 8.0.3317760000.9, 8.0.98344960000.5, 8.0.98344960000.6, 8.0.7965941760000.28, 8.0.7965941760000.34, 8.0.6146560000.1, 8.0.98344960000.4, 8.0.7965941760000.17, 8.0.497871360000.8, 8.0.796594176.1, 8.0.12745506816.5, 8.0.3317760000.8, 8.0.207360000.2, 8.0.12745506816.2, 8.0.12745506816.3, 8.0.3317760000.7, 8.0.3317760000.1, 8.0.98344960000.3, 8.0.6146560000.2, 8.0.7965941760000.16, 8.0.497871360000.1, 8.8.98344960000.1, 8.0.98344960000.1, 8.8.7965941760000.8, 8.0.7965941760000.32, 8.0.12745506816.4, 8.0.796594176.2, 8.0.3317760000.3, 8.0.207360000.1, 8.8.12745506816.1, 8.0.12745506816.6, 8.8.3317760000.1, 8.0.3317760000.6, 8.0.7965941760000.61, 8.0.497871360000.19, 8.0.497871360000.16, 8.0.7965941760000.48, 8.8.7965941760000.2, 8.0.7965941760000.42, 8.0.7965941760000.19, 8.8.7965941760000.1, 8.8.7965941760000.3, 8.0.7965941760000.15, 8.0.7965941760000.45, 8.8.7965941760000.9, 8.0.7965941760000.51, 8.0.497871360000.5, 8.0.497871360000.9, 8.0.7965941760000.36, 8.0.7965941760000.64, 8.0.7965941760000.53, 8.0.7965941760000.43, 8.0.7965941760000.60, 8.0.497871360000.20, 8.0.7965941760000.49, 8.0.497871360000.18, 8.0.7965941760000.9, 8.0.7965941760000.37, 8.0.7965941760000.46, 8.0.7965941760000.56, 8.0.7965941760000.59, 8.0.497871360000.4, 8.0.7965941760000.52, 8.0.497871360000.15, 8.0.7965941760000.31, 8.0.7965941760000.8, 8.0.7965941760000.27, 8.0.7965941760000.14, 8.0.7965941760000.30, 8.0.7965941760000.18, 8.0.7965941760000.5, 8.0.7965941760000.11, 8.0.7965941760000.29, 8.0.7965941760000.2, 8.0.7965941760000.1, 8.0.31116960000.3, 8.0.31116960000.9, 8.0.7965941760000.6, 8.0.7965941760000.7, 8.0.31116960000.6, 8.0.31116960000.5, 8.0.7965941760000.25, 8.0.497871360000.6, 8.0.497871360000.7, 8.0.7965941760000.20, 8.8.497871360000.2, 8.0.7965941760000.23, 8.0.497871360000.2, 8.8.7965941760000.6, 8.8.7965941760000.7, 8.0.7965941760000.22, 8.0.31116960000.1, 8.8.31116960000.1, 8.0.497871360000.12, 8.0.497871360000.11, 8.0.121550625.1, 8.0.31116960000.2, 16.0.63456228123711897600000000.10, 16.0.63456228123711897600000000.12, 16.0.63456228123711897600000000.6, 16.0.9671731157401600000000.1, 16.0.63456228123711897600000000.5, 16.0.162447943996702457856.1, 16.0.11007531417600000000.1, 16.0.63456228123711897600000000.17, 16.0.63456228123711897600000000.7, 16.0.63456228123711897600000000.9, 16.0.63456228123711897600000000.13, 16.0.63456228123711897600000000.3, 16.0.63456228123711897600000000.20, 16.0.63456228123711897600000000.4, 16.0.968265199641600000000.1, 16.0.63456228123711897600000000.21, 16.0.63456228123711897600000000.15, 16.0.63456228123711897600000000.24, 16.0.63456228123711897600000000.2, 16.0.63456228123711897600000000.19, 16.0.63456228123711897600000000.22, 16.0.247875891108249600000000.2, 16.0.63456228123711897600000000.8, 16.0.63456228123711897600000000.1, 16.0.247875891108249600000000.1, 16.16.63456228123711897600000000.1, 16.0.63456228123711897600000000.23, 16.0.63456228123711897600000000.11, 16.0.63456228123711897600000000.18, 16.0.63456228123711897600000000.14, 16.0.63456228123711897600000000.16
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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]
| $p$ |
$2$ |
$3$ |
$5$ |
$7$ |
$11$ |
$13$ |
$17$ |
$19$ |
$23$ |
$29$ |
$31$ |
$37$ |
$41$ |
$43$ |
$47$ |
$53$ |
$59$ |
|---|
| Cycle type |
R |
R |
R |
R |
${\href{/padicField/11.2.0.1}{2} }^{16}$ |
${\href{/padicField/13.2.0.1}{2} }^{16}$ |
${\href{/padicField/17.2.0.1}{2} }^{16}$ |
${\href{/padicField/19.2.0.1}{2} }^{16}$ |
${\href{/padicField/23.2.0.1}{2} }^{16}$ |
${\href{/padicField/29.2.0.1}{2} }^{16}$ |
${\href{/padicField/31.2.0.1}{2} }^{16}$ |
${\href{/padicField/37.2.0.1}{2} }^{16}$ |
${\href{/padicField/41.2.0.1}{2} }^{16}$ |
${\href{/padicField/43.2.0.1}{2} }^{16}$ |
${\href{/padicField/47.2.0.1}{2} }^{16}$ |
${\href{/padicField/53.2.0.1}{2} }^{16}$ |
${\href{/padicField/59.2.0.1}{2} }^{16}$ |
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]
| $p$ | Label | Polynomial
| $e$ |
$f$ |
$c$ |
Galois group |
Slope content |
\(2\)
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| 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | | 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | | 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | | 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
\(3\)
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| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
\(5\)
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| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
\(7\)
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| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
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