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Properties

Label 16.14.734...776.2
Degree $16$
Signature $[14, 1]$
Discriminant $-7.349\times 10^{23}$
Root discriminant \(31.02\)
Ramified primes $2,3,37,4457$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\wr C_2^3.\GL(3,2)$ (as 16T1916)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 22*x^14 - 2*x^13 + 190*x^12 + 30*x^11 - 834*x^10 - 148*x^9 + 2015*x^8 + 278*x^7 - 2670*x^6 - 92*x^5 + 1742*x^4 - 194*x^3 - 388*x^2 + 94*x + 1)
 
gp: K = bnfinit(y^16 - 22*y^14 - 2*y^13 + 190*y^12 + 30*y^11 - 834*y^10 - 148*y^9 + 2015*y^8 + 278*y^7 - 2670*y^6 - 92*y^5 + 1742*y^4 - 194*y^3 - 388*y^2 + 94*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 22*x^14 - 2*x^13 + 190*x^12 + 30*x^11 - 834*x^10 - 148*x^9 + 2015*x^8 + 278*x^7 - 2670*x^6 - 92*x^5 + 1742*x^4 - 194*x^3 - 388*x^2 + 94*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 22*x^14 - 2*x^13 + 190*x^12 + 30*x^11 - 834*x^10 - 148*x^9 + 2015*x^8 + 278*x^7 - 2670*x^6 - 92*x^5 + 1742*x^4 - 194*x^3 - 388*x^2 + 94*x + 1)
 

\( x^{16} - 22 x^{14} - 2 x^{13} + 190 x^{12} + 30 x^{11} - 834 x^{10} - 148 x^{9} + 2015 x^{8} + 278 x^{7} + \cdots + 1 \) 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:   \(-734874894868691892043776\) \(\medspace = -\,2^{24}\cdot 3\cdot 37\cdot 4457^{4}\) 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:  \(31.02\)
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:  $2^{15/8}3^{1/2}37^{1/2}4457^{1/2}\approx 2579.966389949216$
Ramified primes:   \(2\), \(3\), \(37\), \(4457\) 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:  \(\Q(\sqrt{-111}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{9968348558}a^{15}-\frac{87650138}{4984174279}a^{14}+\frac{535719573}{4984174279}a^{13}+\frac{661263191}{9968348558}a^{12}-\frac{2182649703}{9968348558}a^{11}+\frac{36943385}{1424049794}a^{10}-\frac{565269331}{4984174279}a^{9}+\frac{661378303}{4984174279}a^{8}-\frac{852728253}{9968348558}a^{7}+\frac{2225136173}{9968348558}a^{6}-\frac{261538743}{712024897}a^{5}+\frac{1793503575}{4984174279}a^{4}+\frac{1941439939}{4984174279}a^{3}+\frac{1578623043}{9968348558}a^{2}+\frac{1836713418}{4984174279}a+\frac{1812091115}{9968348558}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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:   $\frac{72567233}{1424049794}a^{15}+\frac{61017911}{1424049794}a^{14}-\frac{781174563}{712024897}a^{13}-\frac{687523145}{712024897}a^{12}+\frac{6456524417}{712024897}a^{11}+\frac{5725935549}{712024897}a^{10}-\frac{26301497379}{712024897}a^{9}-\frac{43569044725}{1424049794}a^{8}+\frac{57408413550}{712024897}a^{7}+\frac{38089937072}{712024897}a^{6}-\frac{68767122661}{712024897}a^{5}-\frac{47919578475}{1424049794}a^{4}+\frac{86090774307}{1424049794}a^{3}-\frac{5289728771}{1424049794}a^{2}-\frac{10131962412}{712024897}a+\frac{5498917051}{1424049794}$, $\frac{489978359}{9968348558}a^{15}+\frac{198287811}{4984174279}a^{14}-\frac{10684158601}{9968348558}a^{13}-\frac{9099294383}{9968348558}a^{12}+\frac{45015035519}{4984174279}a^{11}+\frac{5599198497}{712024897}a^{10}-\frac{377583687211}{9968348558}a^{9}-\frac{319287779697}{9968348558}a^{8}+\frac{428845979336}{4984174279}a^{7}+\frac{323962591447}{4984174279}a^{6}-\frac{152923403777}{1424049794}a^{5}-\frac{614834991935}{9968348558}a^{4}+\frac{676615593467}{9968348558}a^{3}+\frac{108451471848}{4984174279}a^{2}-\frac{72841774695}{4984174279}a-\frac{2736392453}{4984174279}$, $\frac{254717545}{9968348558}a^{15}+\frac{112411573}{9968348558}a^{14}-\frac{2527313104}{4984174279}a^{13}-\frac{1741771277}{4984174279}a^{12}+\frac{37622139407}{9968348558}a^{11}+\frac{2578181567}{712024897}a^{10}-\frac{131506120961}{9968348558}a^{9}-\frac{163178974391}{9968348558}a^{8}+\frac{214602831049}{9968348558}a^{7}+\frac{173798270274}{4984174279}a^{6}-\frac{17122745977}{1424049794}a^{5}-\frac{327572052435}{9968348558}a^{4}-\frac{11688633561}{4984174279}a^{3}+\frac{100721333523}{9968348558}a^{2}+\frac{5937837349}{9968348558}a-\frac{11027603367}{9968348558}$, $\frac{154288495}{4984174279}a^{15}+\frac{350665761}{4984174279}a^{14}-\frac{2997126884}{4984174279}a^{13}-\frac{7295049163}{4984174279}a^{12}+\frac{20947325289}{4984174279}a^{11}+\frac{16034715899}{1424049794}a^{10}-\frac{64778676428}{4984174279}a^{9}-\frac{200300160202}{4984174279}a^{8}+\frac{92808557965}{4984174279}a^{7}+\frac{705657206367}{9968348558}a^{6}-\frac{9440036615}{712024897}a^{5}-\frac{299131972171}{4984174279}a^{4}+\frac{34567290343}{4984174279}a^{3}+\frac{203303990383}{9968348558}a^{2}-\frac{15090420918}{4984174279}a-\frac{7983060644}{4984174279}$, $\frac{1251356775}{9968348558}a^{15}+\frac{626634959}{4984174279}a^{14}-\frac{12826810624}{4984174279}a^{13}-\frac{28289091855}{9968348558}a^{12}+\frac{98730760906}{4984174279}a^{11}+\frac{33610818231}{1424049794}a^{10}-\frac{360933726491}{4984174279}a^{9}-\frac{448399484870}{4984174279}a^{8}+\frac{664192548427}{4984174279}a^{7}+\frac{1625352364941}{9968348558}a^{6}-\frac{86205672652}{712024897}a^{5}-\frac{634397961892}{4984174279}a^{4}+\frac{504613392967}{9968348558}a^{3}+\frac{303121450781}{9968348558}a^{2}-\frac{40798226283}{4984174279}a-\frac{13374402817}{9968348558}$, $\frac{130350735}{9968348558}a^{15}+\frac{339814725}{9968348558}a^{14}-\frac{1520846731}{4984174279}a^{13}-\frac{7512153687}{9968348558}a^{12}+\frac{13707315868}{4984174279}a^{11}+\frac{9244000665}{1424049794}a^{10}-\frac{61270714199}{4984174279}a^{9}-\frac{139515631690}{4984174279}a^{8}+\frac{147389675904}{4984174279}a^{7}+\frac{642727537645}{9968348558}a^{6}-\frac{27417089354}{712024897}a^{5}-\frac{378366800135}{4984174279}a^{4}+\frac{251062936745}{9968348558}a^{3}+\frac{181166441891}{4984174279}a^{2}-\frac{30602739856}{4984174279}a-\frac{16837569069}{9968348558}$, $\frac{76912013}{4984174279}a^{15}+\frac{423537461}{9968348558}a^{14}-\frac{1700384662}{4984174279}a^{13}-\frac{4380344602}{4984174279}a^{12}+\frac{27976402233}{9968348558}a^{11}+\frac{4914162715}{712024897}a^{10}-\frac{53593629974}{4984174279}a^{9}-\frac{130160804275}{4984174279}a^{8}+\frac{200890254025}{9968348558}a^{7}+\frac{248257288940}{4984174279}a^{6}-\frac{12701484635}{712024897}a^{5}-\frac{218680369141}{4984174279}a^{4}+\frac{70727743153}{9968348558}a^{3}+\frac{139445326807}{9968348558}a^{2}-\frac{7735987582}{4984174279}a-\frac{2786590843}{4984174279}$, $\frac{1093265615}{9968348558}a^{15}+\frac{368847421}{4984174279}a^{14}-\frac{11781950673}{4984174279}a^{13}-\frac{17620748417}{9968348558}a^{12}+\frac{97916997971}{4984174279}a^{11}+\frac{22296430217}{1424049794}a^{10}-\frac{403231913956}{4984174279}a^{9}-\frac{643144726333}{9968348558}a^{8}+\frac{889871038886}{4984174279}a^{7}+\frac{1295766254945}{9968348558}a^{6}-\frac{150955638058}{712024897}a^{5}-\frac{1195730683897}{9968348558}a^{4}+\frac{1228559527527}{9968348558}a^{3}+\frac{403125551847}{9968348558}a^{2}-\frac{110681337815}{4984174279}a-\frac{208434610}{4984174279}$, $\frac{295497722}{4984174279}a^{15}+\frac{76481026}{4984174279}a^{14}-\frac{6754936963}{4984174279}a^{13}-\frac{4727818553}{9968348558}a^{12}+\frac{60978944839}{4984174279}a^{11}+\frac{3696577919}{712024897}a^{10}-\frac{280549871016}{4984174279}a^{9}-\frac{260302894021}{9968348558}a^{8}+\frac{708444327124}{4984174279}a^{7}+\frac{321375225768}{4984174279}a^{6}-\frac{139162223062}{712024897}a^{5}-\frac{729858934655}{9968348558}a^{4}+\frac{657775293888}{4984174279}a^{3}+\frac{140676392193}{4984174279}a^{2}-\frac{158373852556}{4984174279}a+\frac{3915361272}{4984174279}$, $\frac{241067538}{4984174279}a^{15}+\frac{738791069}{9968348558}a^{14}-\frac{4595593508}{4984174279}a^{13}-\frac{16483328663}{9968348558}a^{12}+\frac{30511806646}{4984174279}a^{11}+\frac{19426947159}{1424049794}a^{10}-\frac{76875893246}{4984174279}a^{9}-\frac{514927018695}{9968348558}a^{8}+\frac{10329339242}{4984174279}a^{7}+\frac{919672244683}{9968348558}a^{6}+\frac{34979497671}{712024897}a^{5}-\frac{697406645057}{9968348558}a^{4}-\frac{309520245096}{4984174279}a^{3}+\frac{90398779897}{4984174279}a^{2}+\frac{77142435940}{4984174279}a-\frac{16022460850}{4984174279}$, $\frac{9062569}{4984174279}a^{15}-\frac{184757747}{4984174279}a^{14}-\frac{404469643}{9968348558}a^{13}+\frac{7813264105}{9968348558}a^{12}+\frac{4606255617}{9968348558}a^{11}-\frac{9182698065}{1424049794}a^{10}-\frac{16300144698}{4984174279}a^{9}+\frac{265923141549}{9968348558}a^{8}+\frac{132097926481}{9968348558}a^{7}-\frac{594469740295}{9968348558}a^{6}-\frac{20373958454}{712024897}a^{5}+\frac{697631841687}{9968348558}a^{4}+\frac{290671148665}{9968348558}a^{3}-\frac{362275656871}{9968348558}a^{2}-\frac{97583689057}{9968348558}a+\frac{24362371379}{4984174279}$, $\frac{401258351}{9968348558}a^{15}-\frac{55365165}{9968348558}a^{14}-\frac{8186492233}{9968348558}a^{13}+\frac{142585508}{4984174279}a^{12}+\frac{63740185819}{9968348558}a^{11}+\frac{319993357}{1424049794}a^{10}-\frac{242460271407}{9968348558}a^{9}-\frac{3390746444}{4984174279}a^{8}+\frac{477647028947}{9968348558}a^{7}-\frac{42045643377}{9968348558}a^{6}-\frac{64690275737}{1424049794}a^{5}+\frac{81539222917}{4984174279}a^{4}+\frac{59831619042}{4984174279}a^{3}-\frac{67114035904}{4984174279}a^{2}+\frac{29554977253}{4984174279}a-\frac{7808387233}{4984174279}$, $\frac{273729363}{9968348558}a^{15}+\frac{953237293}{9968348558}a^{14}-\frac{5288092203}{9968348558}a^{13}-\frac{10019393483}{4984174279}a^{12}+\frac{17710265947}{4984174279}a^{11}+\frac{22573811573}{1424049794}a^{10}-\frac{93249451573}{9968348558}a^{9}-\frac{294583730493}{4984174279}a^{8}+\frac{34868876083}{4984174279}a^{7}+\frac{1097207389073}{9968348558}a^{6}+\frac{1219649431}{1424049794}a^{5}-\frac{494665906895}{4984174279}a^{4}+\frac{141410613289}{9968348558}a^{3}+\frac{187165967180}{4984174279}a^{2}-\frac{92572685375}{4984174279}a-\frac{252276796}{4984174279}$, $\frac{105034725}{9968348558}a^{15}+\frac{43093082}{4984174279}a^{14}-\frac{1466926210}{4984174279}a^{13}-\frac{1283059429}{4984174279}a^{12}+\frac{32575613251}{9968348558}a^{11}+\frac{2049219988}{712024897}a^{10}-\frac{182908902685}{9968348558}a^{9}-\frac{151372365115}{9968348558}a^{8}+\frac{551011597491}{9968348558}a^{7}+\frac{192633932840}{4984174279}a^{6}-\frac{126644599063}{1424049794}a^{5}-\frac{416972839687}{9968348558}a^{4}+\frac{350162361483}{4984174279}a^{3}+\frac{40817842971}{4984174279}a^{2}-\frac{158387261491}{9968348558}a+\frac{24475706377}{9968348558}$ 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:  \( 7012404.18887 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{1}\cdot 7012404.18887 \cdot 1}{2\cdot\sqrt{734874894868691892043776}}\cr\approx \mathstrut & 0.421046485822 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 22*x^14 - 2*x^13 + 190*x^12 + 30*x^11 - 834*x^10 - 148*x^9 + 2015*x^8 + 278*x^7 - 2670*x^6 - 92*x^5 + 1742*x^4 - 194*x^3 - 388*x^2 + 94*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 - 22*x^14 - 2*x^13 + 190*x^12 + 30*x^11 - 834*x^10 - 148*x^9 + 2015*x^8 + 278*x^7 - 2670*x^6 - 92*x^5 + 1742*x^4 - 194*x^3 - 388*x^2 + 94*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 - 22*x^14 - 2*x^13 + 190*x^12 + 30*x^11 - 834*x^10 - 148*x^9 + 2015*x^8 + 278*x^7 - 2670*x^6 - 92*x^5 + 1742*x^4 - 194*x^3 - 388*x^2 + 94*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 - 22*x^14 - 2*x^13 + 190*x^12 + 30*x^11 - 834*x^10 - 148*x^9 + 2015*x^8 + 278*x^7 - 2670*x^6 - 92*x^5 + 1742*x^4 - 194*x^3 - 388*x^2 + 94*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

$C_2\wr C_2^3.\GL(3,2)$ (as 16T1916):

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 non-solvable group of order 344064
The 79 conjugacy class representatives for $C_2\wr C_2^3.\GL(3,2)$
Character table for $C_2\wr C_2^3.\GL(3,2)$

Intermediate fields

8.8.81366421504.1

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]
 

Sibling fields

Degree 16 sibling: data not computed
Degree 32 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ R ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 2.8.12.17$x^{8} + 4 x^{7} + 6 x^{6} + 4 x^{5} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 4$$4$$2$$12$$C_2^4:C_6$$[2, 2, 2, 2]^{6}$
2.8.12.17$x^{8} + 4 x^{7} + 6 x^{6} + 4 x^{5} + 8 x^{4} + 8 x^{3} + 4 x^{2} + 4$$4$$2$$12$$C_2^4:C_6$$[2, 2, 2, 2]^{6}$
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.7.0.1$x^{7} + 2 x^{2} + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
3.7.0.1$x^{7} + 2 x^{2} + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.3.0.1$x^{3} + 6 x + 35$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} + 6 x + 35$$1$$3$$0$$C_3$$[\ ]^{3}$
37.6.0.1$x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
\(4457\) Copy content Toggle raw display $\Q_{4457}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{4457}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$