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Properties

Label 16.14.481...000.1
Degree $16$
Signature $[14, 1]$
Discriminant $-4.810\times 10^{21}$
Root discriminant \(22.65\)
Ramified primes $2,3,5,179$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\wr (C_2\times C_4)$ (as 16T1379)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 2*x^14 + 76*x^13 - 118*x^12 - 194*x^11 + 618*x^10 - 346*x^9 - 266*x^8 + 350*x^7 - 216*x^6 + 62*x^5 + 123*x^4 - 70*x^3 - 20*x^2 + 8*x + 1)
 
gp: K = bnfinit(y^16 - 6*y^15 - 2*y^14 + 76*y^13 - 118*y^12 - 194*y^11 + 618*y^10 - 346*y^9 - 266*y^8 + 350*y^7 - 216*y^6 + 62*y^5 + 123*y^4 - 70*y^3 - 20*y^2 + 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 2*x^14 + 76*x^13 - 118*x^12 - 194*x^11 + 618*x^10 - 346*x^9 - 266*x^8 + 350*x^7 - 216*x^6 + 62*x^5 + 123*x^4 - 70*x^3 - 20*x^2 + 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 - 2*x^14 + 76*x^13 - 118*x^12 - 194*x^11 + 618*x^10 - 346*x^9 - 266*x^8 + 350*x^7 - 216*x^6 + 62*x^5 + 123*x^4 - 70*x^3 - 20*x^2 + 8*x + 1)
 

\( x^{16} - 6 x^{15} - 2 x^{14} + 76 x^{13} - 118 x^{12} - 194 x^{11} + 618 x^{10} - 346 x^{9} - 266 x^{8} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[14, 1]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-4810420224000000000000\) \(\medspace = -\,2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 179\) 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:  \(22.65\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(5\), \(179\) 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:  \(\Q(\sqrt{-179}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{126249537}a^{15}+\frac{17422016}{126249537}a^{14}+\frac{7922153}{126249537}a^{13}+\frac{16269395}{126249537}a^{12}-\frac{23724658}{126249537}a^{11}-\frac{40279124}{126249537}a^{10}+\frac{748709}{126249537}a^{9}+\frac{6672770}{126249537}a^{8}+\frac{4997155}{126249537}a^{7}+\frac{41961572}{126249537}a^{6}-\frac{46349173}{126249537}a^{5}-\frac{552065}{3079257}a^{4}+\frac{143408}{1026419}a^{3}-\frac{58778572}{126249537}a^{2}-\frac{59604533}{126249537}a-\frac{16030129}{42083179}$ 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

Trivial group, which has order $1$ (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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{266646}{1026419}a^{15}-\frac{1983362}{1026419}a^{14}+\frac{1102916}{1026419}a^{13}+\frac{23565536}{1026419}a^{12}-\frac{53417754}{1026419}a^{11}-\frac{42641446}{1026419}a^{10}+\frac{235788046}{1026419}a^{9}-\frac{189002025}{1026419}a^{8}-\frac{50387726}{1026419}a^{7}+\frac{117620692}{1026419}a^{6}-\frac{100443978}{1026419}a^{5}+\frac{57878517}{1026419}a^{4}+\frac{31386270}{1026419}a^{3}-\frac{26596028}{1026419}a^{2}-\frac{2433948}{1026419}a+\frac{714462}{1026419}$, $\frac{20799764}{126249537}a^{15}-\frac{56514160}{126249537}a^{14}-\frac{341676527}{126249537}a^{13}+\frac{344287341}{42083179}a^{12}+\frac{1542484553}{126249537}a^{11}-\frac{6239537791}{126249537}a^{10}+\frac{258448442}{126249537}a^{9}+\frac{12490178104}{126249537}a^{8}-\frac{10983689597}{126249537}a^{7}+\frac{727987090}{126249537}a^{6}+\frac{1444549826}{42083179}a^{5}-\frac{148162961}{3079257}a^{4}+\frac{28647181}{1026419}a^{3}+\frac{900521368}{126249537}a^{2}-\frac{1044682184}{126249537}a+\frac{22968494}{42083179}$, $\frac{8705415}{42083179}a^{15}-\frac{180201631}{126249537}a^{14}+\frac{10395548}{42083179}a^{13}+\frac{2226133306}{126249537}a^{12}-\frac{4170217489}{126249537}a^{11}-\frac{1699523524}{42083179}a^{10}+\frac{19339732664}{126249537}a^{9}-\frac{4113652389}{42083179}a^{8}-\frac{4376254355}{126249537}a^{7}+\frac{2572122938}{42083179}a^{6}-\frac{8353443329}{126249537}a^{5}+\frac{93039901}{3079257}a^{4}+\frac{19230620}{1026419}a^{3}-\frac{300581072}{42083179}a^{2}+\frac{20142097}{126249537}a-\frac{29165643}{42083179}$, $\frac{33878932}{126249537}a^{15}-\frac{183357464}{126249537}a^{14}-\frac{48197712}{42083179}a^{13}+\frac{2386194289}{126249537}a^{12}-\frac{2984861267}{126249537}a^{11}-\frac{6781660030}{126249537}a^{10}+\frac{17836757150}{126249537}a^{9}-\frac{7883160680}{126249537}a^{8}-\frac{10687147982}{126249537}a^{7}+\frac{3937076832}{42083179}a^{6}-\frac{5646065219}{126249537}a^{5}+\frac{21079973}{3079257}a^{4}+\frac{113998247}{3079257}a^{3}-\frac{856576483}{42083179}a^{2}-\frac{243942834}{42083179}a+\frac{27151846}{42083179}$, $\frac{30472405}{126249537}a^{15}-\frac{190573924}{126249537}a^{14}+\frac{12509711}{126249537}a^{13}+\frac{747506167}{42083179}a^{12}-\frac{1498335558}{42083179}a^{11}-\frac{3780859220}{126249537}a^{10}+\frac{6926100837}{42083179}a^{9}-\frac{6511577759}{42083179}a^{8}-\frac{827957003}{42083179}a^{7}+\frac{12769858426}{126249537}a^{6}-\frac{3404948117}{42083179}a^{5}+\frac{50456560}{1026419}a^{4}+\frac{14948673}{1026419}a^{3}-\frac{970541440}{42083179}a^{2}+\frac{81705146}{126249537}a+\frac{178582540}{126249537}$, $\frac{46916009}{126249537}a^{15}-\frac{236715791}{126249537}a^{14}-\frac{310489883}{126249537}a^{13}+\frac{3258995329}{126249537}a^{12}-\frac{2627732936}{126249537}a^{11}-\frac{11338108363}{126249537}a^{10}+\frac{19598181106}{126249537}a^{9}+\frac{149220937}{126249537}a^{8}-\frac{15359943952}{126249537}a^{7}+\frac{8444355904}{126249537}a^{6}-\frac{4019793851}{126249537}a^{5}-\frac{55123060}{3079257}a^{4}+\frac{47877801}{1026419}a^{3}-\frac{1221848}{126249537}a^{2}-\frac{1024540087}{126249537}a-\frac{48280328}{42083179}$, $\frac{53597222}{126249537}a^{15}-\frac{300467686}{126249537}a^{14}-\frac{206017859}{126249537}a^{13}+\frac{1310474317}{42083179}a^{12}-\frac{5027899189}{126249537}a^{11}-\frac{11484435649}{126249537}a^{10}+\frac{29260378100}{126249537}a^{9}-\frac{10757070971}{126249537}a^{8}-\frac{17181379895}{126249537}a^{7}+\frac{15195332206}{126249537}a^{6}-\frac{2673653272}{42083179}a^{5}+\frac{25472590}{3079257}a^{4}+\frac{60033451}{1026419}a^{3}-\frac{2370790076}{126249537}a^{2}-\frac{1344057788}{126249537}a+\frac{52261436}{42083179}$, $\frac{8027434}{126249537}a^{15}-\frac{103520176}{126249537}a^{14}+\frac{61883647}{42083179}a^{13}+\frac{1230204694}{126249537}a^{12}-\frac{1271286106}{42083179}a^{11}-\frac{2171936992}{126249537}a^{10}+\frac{15968232304}{126249537}a^{9}-\frac{10537747288}{126249537}a^{8}-\frac{7855074688}{126249537}a^{7}+\frac{8343806267}{126249537}a^{6}-\frac{4806266447}{126249537}a^{5}+\frac{17205907}{1026419}a^{4}+\frac{98212883}{3079257}a^{3}-\frac{1811072792}{126249537}a^{2}-\frac{300518647}{42083179}a+\frac{186156859}{126249537}$, $\frac{8705415}{42083179}a^{15}-\frac{180201631}{126249537}a^{14}+\frac{10395548}{42083179}a^{13}+\frac{2226133306}{126249537}a^{12}-\frac{4170217489}{126249537}a^{11}-\frac{1699523524}{42083179}a^{10}+\frac{19339732664}{126249537}a^{9}-\frac{4113652389}{42083179}a^{8}-\frac{4376254355}{126249537}a^{7}+\frac{2572122938}{42083179}a^{6}-\frac{8353443329}{126249537}a^{5}+\frac{93039901}{3079257}a^{4}+\frac{19230620}{1026419}a^{3}-\frac{300581072}{42083179}a^{2}+\frac{20142097}{126249537}a-\frac{71248822}{42083179}$, $\frac{38211899}{126249537}a^{15}-\frac{264028445}{126249537}a^{14}+\frac{28631330}{42083179}a^{13}+\frac{3141781385}{126249537}a^{12}-\frac{2208598008}{42083179}a^{11}-\frac{5727215441}{126249537}a^{10}+\frac{29769583658}{126249537}a^{9}-\frac{25156521998}{126249537}a^{8}-\frac{4588387685}{126249537}a^{7}+\frac{16402663297}{126249537}a^{6}-\frac{14531712121}{126249537}a^{5}+\frac{65980292}{1026419}a^{4}+\frac{63835189}{3079257}a^{3}-\frac{3602777566}{126249537}a^{2}+\frac{48967282}{42083179}a+\frac{209054945}{126249537}$, $\frac{17256635}{42083179}a^{15}-\frac{100104623}{42083179}a^{14}-\frac{171877613}{126249537}a^{13}+\frac{3914498854}{126249537}a^{12}-\frac{1725037046}{42083179}a^{11}-\frac{11310646403}{126249537}a^{10}+\frac{28792928668}{126249537}a^{9}-\frac{10884928780}{126249537}a^{8}-\frac{13656163858}{126249537}a^{7}+\frac{12896110660}{126249537}a^{6}-\frac{10331921807}{126249537}a^{5}+\frac{18177773}{1026419}a^{4}+\frac{144413579}{3079257}a^{3}-\frac{1407210638}{126249537}a^{2}-\frac{232783832}{126249537}a-\frac{27663516}{42083179}$, $\frac{16973589}{42083179}a^{15}-\frac{285806312}{126249537}a^{14}-\frac{218951756}{126249537}a^{13}+\frac{3808553569}{126249537}a^{12}-\frac{4463421844}{126249537}a^{11}-\frac{11921666720}{126249537}a^{10}+\frac{8922086129}{42083179}a^{9}-\frac{2064988144}{42083179}a^{8}-\frac{5472944771}{42083179}a^{7}+\frac{3776012179}{42083179}a^{6}-\frac{7614783962}{126249537}a^{5}+\frac{19077877}{3079257}a^{4}+\frac{157154918}{3079257}a^{3}-\frac{854267344}{126249537}a^{2}-\frac{819475967}{126249537}a-\frac{204863150}{126249537}$, $\frac{6041103}{42083179}a^{15}-\frac{41852786}{126249537}a^{14}-\frac{354610424}{126249537}a^{13}+\frac{909992641}{126249537}a^{12}+\frac{2106961898}{126249537}a^{11}-\frac{6676768862}{126249537}a^{10}-\frac{745223757}{42083179}a^{9}+\frac{5684094881}{42083179}a^{8}-\frac{3407048005}{42083179}a^{7}-\frac{1046436193}{42083179}a^{6}+\frac{4739825332}{126249537}a^{5}-\frac{154557674}{3079257}a^{4}+\frac{62996108}{3079257}a^{3}+\frac{2417044100}{126249537}a^{2}-\frac{520100363}{126249537}a-\frac{166492439}{126249537}$, $\frac{4895734}{42083179}a^{15}+\frac{130229}{42083179}a^{14}-\frac{113967394}{42083179}a^{13}+\frac{65433754}{126249537}a^{12}+\frac{2823515614}{126249537}a^{11}-\frac{370456687}{42083179}a^{10}-\frac{9573481652}{126249537}a^{9}+\frac{5306118598}{126249537}a^{8}+\frac{10629449585}{126249537}a^{7}-\frac{6368933704}{126249537}a^{6}+\frac{656693047}{126249537}a^{5}-\frac{1930339}{3079257}a^{4}-\frac{40874875}{1026419}a^{3}+\frac{1742103172}{126249537}a^{2}+\frac{1246565158}{126249537}a-\frac{51334535}{126249537}$ 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:  \( 205339.151919 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{1}\cdot 205339.151919 \cdot 1}{2\cdot\sqrt{4810420224000000000000}}\cr\approx \mathstrut & 0.152387753813 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 2*x^14 + 76*x^13 - 118*x^12 - 194*x^11 + 618*x^10 - 346*x^9 - 266*x^8 + 350*x^7 - 216*x^6 + 62*x^5 + 123*x^4 - 70*x^3 - 20*x^2 + 8*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^16 - 6*x^15 - 2*x^14 + 76*x^13 - 118*x^12 - 194*x^11 + 618*x^10 - 346*x^9 - 266*x^8 + 350*x^7 - 216*x^6 + 62*x^5 + 123*x^4 - 70*x^3 - 20*x^2 + 8*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^16 - 6*x^15 - 2*x^14 + 76*x^13 - 118*x^12 - 194*x^11 + 618*x^10 - 346*x^9 - 266*x^8 + 350*x^7 - 216*x^6 + 62*x^5 + 123*x^4 - 70*x^3 - 20*x^2 + 8*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^16 - 6*x^15 - 2*x^14 + 76*x^13 - 118*x^12 - 194*x^11 + 618*x^10 - 346*x^9 - 266*x^8 + 350*x^7 - 216*x^6 + 62*x^5 + 123*x^4 - 70*x^3 - 20*x^2 + 8*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

$C_2\wr (C_2\times C_4)$ (as 16T1379):

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 2048
The 71 conjugacy class representatives for $C_2\wr (C_2\times C_4)$
Character table for $C_2\wr (C_2\times C_4)$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.12.551081740861440000000000.1

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 ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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
\(2\) Copy content Toggle raw display Deg $16$$4$$4$$24$
\(3\) Copy content Toggle raw display 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}$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(179\) Copy content Toggle raw display $\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
179.2.1.2$x^{2} + 179$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$