↑

Properties

Label 32.0.125...625.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.250\times 10^{48}$
Root discriminant \(31.84\)
Ramified primes $5,17$
Class number $17$ (GRH)
Class group [17] (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 2*x^30 - 3*x^29 + 5*x^28 - 8*x^27 + 13*x^26 - 21*x^25 + 34*x^24 - 55*x^23 + 89*x^22 - 144*x^21 + 233*x^20 - 377*x^19 + 610*x^18 - 987*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 
gp: K = bnfinit(y^32 - y^31 + 2*y^30 - 3*y^29 + 5*y^28 - 8*y^27 + 13*y^26 - 21*y^25 + 34*y^24 - 55*y^23 + 89*y^22 - 144*y^21 + 233*y^20 - 377*y^19 + 610*y^18 - 987*y^17 + 1597*y^16 + 987*y^15 + 610*y^14 + 377*y^13 + 233*y^12 + 144*y^11 + 89*y^10 + 55*y^9 + 34*y^8 + 21*y^7 + 13*y^6 + 8*y^5 + 5*y^4 + 3*y^3 + 2*y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - x^31 + 2*x^30 - 3*x^29 + 5*x^28 - 8*x^27 + 13*x^26 - 21*x^25 + 34*x^24 - 55*x^23 + 89*x^22 - 144*x^21 + 233*x^20 - 377*x^19 + 610*x^18 - 987*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - x^31 + 2*x^30 - 3*x^29 + 5*x^28 - 8*x^27 + 13*x^26 - 21*x^25 + 34*x^24 - 55*x^23 + 89*x^22 - 144*x^21 + 233*x^20 - 377*x^19 + 610*x^18 - 987*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 

\( x^{32} - x^{31} + 2 x^{30} - 3 x^{29} + 5 x^{28} - 8 x^{27} + 13 x^{26} - 21 x^{25} + 34 x^{24} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1250223652010309685753479160887484565582275390625\) \(\medspace = 5^{16}\cdot 17^{30}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(31.84\)
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:  $5^{1/2}17^{15/16}\approx 31.844238354614745$
Ramified primes:   \(5\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(85=5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{85}(1,·)$, $\chi_{85}(4,·)$, $\chi_{85}(6,·)$, $\chi_{85}(9,·)$, $\chi_{85}(11,·)$, $\chi_{85}(14,·)$, $\chi_{85}(16,·)$, $\chi_{85}(19,·)$, $\chi_{85}(21,·)$, $\chi_{85}(24,·)$, $\chi_{85}(26,·)$, $\chi_{85}(29,·)$, $\chi_{85}(31,·)$, $\chi_{85}(36,·)$, $\chi_{85}(39,·)$, $\chi_{85}(41,·)$, $\chi_{85}(44,·)$, $\chi_{85}(46,·)$, $\chi_{85}(49,·)$, $\chi_{85}(54,·)$, $\chi_{85}(56,·)$, $\chi_{85}(59,·)$, $\chi_{85}(61,·)$, $\chi_{85}(64,·)$, $\chi_{85}(66,·)$, $\chi_{85}(69,·)$, $\chi_{85}(71,·)$, $\chi_{85}(74,·)$, $\chi_{85}(76,·)$, $\chi_{85}(79,·)$, $\chi_{85}(81,·)$, $\chi_{85}(84,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1597}a^{17}-\frac{610}{1597}$, $\frac{1}{1597}a^{18}-\frac{610}{1597}a$, $\frac{1}{1597}a^{19}-\frac{610}{1597}a^{2}$, $\frac{1}{1597}a^{20}-\frac{610}{1597}a^{3}$, $\frac{1}{1597}a^{21}-\frac{610}{1597}a^{4}$, $\frac{1}{1597}a^{22}-\frac{610}{1597}a^{5}$, $\frac{1}{1597}a^{23}-\frac{610}{1597}a^{6}$, $\frac{1}{1597}a^{24}-\frac{610}{1597}a^{7}$, $\frac{1}{1597}a^{25}-\frac{610}{1597}a^{8}$, $\frac{1}{1597}a^{26}-\frac{610}{1597}a^{9}$, $\frac{1}{1597}a^{27}-\frac{610}{1597}a^{10}$, $\frac{1}{1597}a^{28}-\frac{610}{1597}a^{11}$, $\frac{1}{1597}a^{29}-\frac{610}{1597}a^{12}$, $\frac{1}{1597}a^{30}-\frac{610}{1597}a^{13}$, $\frac{1}{1597}a^{31}-\frac{610}{1597}a^{14}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{17}$, which has order $17$ (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:   \( -\frac{13}{1597} a^{24} - \frac{46368}{1597} a^{7} \)  (order $34$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{13}{1597}a^{25}+\frac{46368}{1597}a^{8}$, $\frac{2}{1597}a^{20}+\frac{6765}{1597}a^{3}+1$, $\frac{610}{1597}a^{31}-\frac{1220}{1597}a^{30}+\frac{1830}{1597}a^{29}-\frac{2961}{1597}a^{28}+\frac{4880}{1597}a^{27}-\frac{7930}{1597}a^{26}+\frac{12810}{1597}a^{25}-\frac{20740}{1597}a^{24}+\frac{33550}{1597}a^{23}-\frac{54290}{1597}a^{22}+\frac{87840}{1597}a^{21}-\frac{142130}{1597}a^{20}+\frac{229970}{1597}a^{19}-\frac{372100}{1597}a^{18}+\frac{602070}{1597}a^{17}-610a^{16}+987a^{15}-\frac{372100}{1597}a^{14}-\frac{229970}{1597}a^{13}-\frac{142130}{1597}a^{12}+\frac{229971}{1597}a^{11}-\frac{54290}{1597}a^{10}-\frac{33550}{1597}a^{9}-\frac{20740}{1597}a^{8}-\frac{12810}{1597}a^{7}-\frac{7930}{1597}a^{6}-\frac{4880}{1597}a^{5}-\frac{3050}{1597}a^{4}-\frac{1830}{1597}a^{3}-\frac{1220}{1597}a^{2}-\frac{610}{1597}a-\frac{610}{1597}$, $\frac{8}{1597}a^{23}+\frac{2}{1597}a^{20}+\frac{28657}{1597}a^{6}+\frac{6765}{1597}a^{3}+1$, $\frac{610}{1597}a^{31}-\frac{1220}{1597}a^{30}+\frac{1830}{1597}a^{29}-\frac{3050}{1597}a^{28}+\frac{4880}{1597}a^{27}-\frac{7930}{1597}a^{26}+\frac{12810}{1597}a^{25}-\frac{20740}{1597}a^{24}+\frac{33550}{1597}a^{23}-\frac{54290}{1597}a^{22}+\frac{87840}{1597}a^{21}-\frac{142130}{1597}a^{20}+\frac{229970}{1597}a^{19}-\frac{372100}{1597}a^{18}+\frac{602070}{1597}a^{17}-610a^{16}+987a^{15}-\frac{372100}{1597}a^{14}-\frac{229970}{1597}a^{13}-\frac{142130}{1597}a^{12}-\frac{87840}{1597}a^{11}-\frac{54290}{1597}a^{10}-\frac{33550}{1597}a^{9}-\frac{20740}{1597}a^{8}-\frac{12810}{1597}a^{7}-\frac{7930}{1597}a^{6}-\frac{4880}{1597}a^{5}-\frac{3050}{1597}a^{4}-\frac{1830}{1597}a^{3}-\frac{1220}{1597}a^{2}-\frac{610}{1597}a+\frac{987}{1597}$, $\frac{21}{1597}a^{25}+\frac{1}{1597}a^{19}+\frac{75025}{1597}a^{8}+\frac{4181}{1597}a^{2}$, $\frac{21}{1597}a^{25}+\frac{1}{1597}a^{18}+\frac{75025}{1597}a^{8}+\frac{2584}{1597}a$, $\frac{21}{1597}a^{25}+\frac{75025}{1597}a^{8}+1$, $\frac{55}{1597}a^{27}-\frac{5}{1597}a^{23}+\frac{1}{1597}a^{19}+\frac{196418}{1597}a^{10}-\frac{17711}{1597}a^{6}+\frac{4181}{1597}a^{2}$, $\frac{1220}{1597}a^{31}-a^{30}+2a^{29}-3a^{28}+5a^{27}-8a^{26}+13a^{25}-21a^{24}+34a^{23}-55a^{22}+89a^{21}-144a^{20}+233a^{19}-\frac{602070}{1597}a^{18}+610a^{17}-987a^{16}+1597a^{15}+\frac{229970}{1597}a^{14}+610a^{13}+377a^{12}+233a^{11}+144a^{10}+89a^{9}+55a^{8}+34a^{7}+21a^{6}+13a^{5}+8a^{4}+5a^{3}+3a^{2}+\frac{610}{1597}a+1$, $\frac{987}{1597}a^{31}-\frac{1220}{1597}a^{30}+\frac{1830}{1597}a^{29}-\frac{3050}{1597}a^{28}+\frac{4880}{1597}a^{27}-\frac{7930}{1597}a^{26}+\frac{12810}{1597}a^{25}-\frac{20740}{1597}a^{24}+\frac{33563}{1597}a^{23}-\frac{54290}{1597}a^{22}+\frac{87840}{1597}a^{21}-\frac{142130}{1597}a^{20}+\frac{229970}{1597}a^{19}-\frac{372100}{1597}a^{18}+\frac{602070}{1597}a^{17}-610a^{16}+987a^{15}+\frac{974169}{1597}a^{14}-\frac{229970}{1597}a^{13}-\frac{142130}{1597}a^{12}-\frac{87840}{1597}a^{11}-\frac{54290}{1597}a^{10}-\frac{33550}{1597}a^{9}-\frac{20740}{1597}a^{8}-\frac{12810}{1597}a^{7}+\frac{38438}{1597}a^{6}-\frac{4880}{1597}a^{5}-\frac{3050}{1597}a^{4}-\frac{1830}{1597}a^{3}-\frac{1220}{1597}a^{2}-\frac{610}{1597}a-\frac{610}{1597}$, $\frac{34}{1597}a^{26}+\frac{34}{1597}a^{25}+\frac{13}{1597}a^{24}+\frac{121393}{1597}a^{9}+\frac{121393}{1597}a^{8}+\frac{46368}{1597}a^{7}$, $\frac{987}{1597}a^{31}-\frac{987}{1597}a^{30}+\frac{1974}{1597}a^{29}-\frac{3016}{1597}a^{28}+\frac{4935}{1597}a^{27}-\frac{7896}{1597}a^{26}+\frac{12831}{1597}a^{25}-\frac{20727}{1597}a^{24}+\frac{33550}{1597}a^{23}-\frac{54285}{1597}a^{22}+\frac{87843}{1597}a^{21}-\frac{142128}{1597}a^{20}+\frac{229971}{1597}a^{19}-\frac{372099}{1597}a^{18}+\frac{602070}{1597}a^{17}-610a^{16}+987a^{15}+\frac{974169}{1597}a^{14}+\frac{602070}{1597}a^{13}+\frac{372099}{1597}a^{12}+\frac{33553}{1597}a^{11}+\frac{142128}{1597}a^{10}+\frac{87843}{1597}a^{9}+\frac{54285}{1597}a^{8}+\frac{33558}{1597}a^{7}-\frac{7930}{1597}a^{6}+\frac{12831}{1597}a^{5}+\frac{7896}{1597}a^{4}+\frac{4935}{1597}a^{3}+\frac{2961}{1597}a^{2}+\frac{1974}{1597}a+\frac{987}{1597}$, $\frac{1364}{1597}a^{31}-a^{30}+\frac{2495}{1597}a^{29}-\frac{3715}{1597}a^{28}+\frac{6066}{1597}a^{27}-\frac{9781}{1597}a^{26}+\frac{15847}{1597}a^{25}-\frac{25615}{1597}a^{24}+\frac{41467}{1597}a^{23}-\frac{67098}{1597}a^{22}+\frac{108580}{1597}a^{21}-\frac{175682}{1597}a^{20}+\frac{284260}{1597}a^{19}-\frac{459940}{1597}a^{18}+\frac{744200}{1597}a^{17}-754a^{16}+1220a^{15}+\frac{1718369}{1597}a^{14}-377a^{13}+\frac{656358}{1597}a^{12}+\frac{87842}{1597}a^{11}+\frac{54287}{1597}a^{10}+\frac{33555}{1597}a^{9}+\frac{20732}{1597}a^{8}+\frac{59191}{1597}a^{7}-\frac{20748}{1597}a^{6}+\frac{22625}{1597}a^{5}+\frac{9760}{1597}a^{4}-\frac{2262}{1597}a^{3}+\frac{3660}{1597}a^{2}-\frac{754}{1597}a+\frac{1220}{1597}$, $\frac{233}{1597}a^{31}-\frac{89}{1597}a^{30}+\frac{34}{1597}a^{27}-\frac{13}{1597}a^{26}+\frac{13}{1597}a^{25}+\frac{5}{1597}a^{23}-\frac{2}{1597}a^{22}+\frac{2}{1597}a^{21}-\frac{1}{1597}a^{18}+\frac{1}{1597}a^{17}+\frac{832040}{1597}a^{14}-\frac{317811}{1597}a^{13}+\frac{121393}{1597}a^{10}-\frac{46368}{1597}a^{9}+\frac{46368}{1597}a^{8}+\frac{17711}{1597}a^{6}-\frac{6765}{1597}a^{5}+\frac{6765}{1597}a^{4}-\frac{987}{1597}a+\frac{987}{1597}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 42597284566.379654 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 42597284566.379654 \cdot 17}{34\cdot\sqrt{1250223652010309685753479160887484565582275390625}}\cr\approx \mathstrut & 0.112392144505091 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 2*x^30 - 3*x^29 + 5*x^28 - 8*x^27 + 13*x^26 - 21*x^25 + 34*x^24 - 55*x^23 + 89*x^22 - 144*x^21 + 233*x^20 - 377*x^19 + 610*x^18 - 987*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^32 - x^31 + 2*x^30 - 3*x^29 + 5*x^28 - 8*x^27 + 13*x^26 - 21*x^25 + 34*x^24 - 55*x^23 + 89*x^22 - 144*x^21 + 233*x^20 - 377*x^19 + 610*x^18 - 987*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - x^31 + 2*x^30 - 3*x^29 + 5*x^28 - 8*x^27 + 13*x^26 - 21*x^25 + 34*x^24 - 55*x^23 + 89*x^22 - 144*x^21 + 233*x^20 - 377*x^19 + 610*x^18 - 987*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - x^31 + 2*x^30 - 3*x^29 + 5*x^28 - 8*x^27 + 13*x^26 - 21*x^25 + 34*x^24 - 55*x^23 + 89*x^22 - 144*x^21 + 233*x^20 - 377*x^19 + 610*x^18 - 987*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\times C_{16}$ (as 32T32):

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)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_{16}$
Character table for $C_2\times C_{16}$

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.4913.1, 4.4.122825.1, 8.8.15085980625.1, \(\Q(\zeta_{17})^+\), 8.8.256461670625.1, 16.16.65772588499765987890625.1, \(\Q(\zeta_{17})\), 16.0.1118134004496021794140625.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{4}$ $16^{2}$ R $16^{2}$ $16^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ R ${\href{/padicField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{4}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $32$$2$$16$$16$
\(17\) Copy content Toggle raw display Deg $32$$16$$2$$30$