LMFDB - Number field 14.10.3161612016871142273.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 6*x^12 + 28*x^11 + 29*x^10 - 61*x^9 - 215*x^8 + 237*x^7 + 347*x^6 - 324*x^5 - 202*x^4 + 139*x^3 + 47*x^2 - 14*x - 1)

gp: K = bnfinit(y^14 - 4*y^13 - 6*y^12 + 28*y^11 + 29*y^10 - 61*y^9 - 215*y^8 + 237*y^7 + 347*y^6 - 324*y^5 - 202*y^4 + 139*y^3 + 47*y^2 - 14*y - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 4*x^13 - 6*x^12 + 28*x^11 + 29*x^10 - 61*x^9 - 215*x^8 + 237*x^7 + 347*x^6 - 324*x^5 - 202*x^4 + 139*x^3 + 47*x^2 - 14*x - 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 4*x^13 - 6*x^12 + 28*x^11 + 29*x^10 - 61*x^9 - 215*x^8 + 237*x^7 + 347*x^6 - 324*x^5 - 202*x^4 + 139*x^3 + 47*x^2 - 14*x - 1)

$ x^{14} - 4 x^{13} - 6 x^{12} + 28 x^{11} + 29 x^{10} - 61 x^{9} - 215 x^{8} + 237 x^{7} + 347 x^{6} + \cdots - 1 $ x^14 - 4*x^13 - 6*x^12 + 28*x^11 + 29*x^10 - 61*x^9 - 215*x^8 + 237*x^7 + 347*x^6 - 324*x^5 - 202*x^4 + 139*x^3 + 47*x^2 - 14*x - 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$14$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[10, 2]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$3161612016871142273$ 3161612016871142273 $\medspace = 17^{5}\cdot 23^{2}\cdot 64879^{2}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$20.96$	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:	$17^{1/2}23^{1/2}64879^{1/2}\approx 5036.634689949233$
Ramified primes:	$17$, $23$, $64879$ 17, 23, 64879	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{17}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{10471571237}a^{13}-\frac{4081298086}{10471571237}a^{12}+\frac{2610428045}{10471571237}a^{11}-\frac{3747178800}{10471571237}a^{10}-\frac{3448605331}{10471571237}a^{9}+\frac{1092756058}{10471571237}a^{8}+\frac{2672491688}{10471571237}a^{7}-\frac{1444137529}{10471571237}a^{6}+\frac{1986766771}{10471571237}a^{5}+\frac{3287951789}{10471571237}a^{4}-\frac{1390306656}{10471571237}a^{3}+\frac{282911940}{10471571237}a^{2}+\frac{1424474522}{10471571237}a-\frac{2272174995}{10471571237}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, 1/10471571237*a^13 - 4081298086/10471571237*a^12 + 2610428045/10471571237*a^11 - 3747178800/10471571237*a^10 - 3448605331/10471571237*a^9 + 1092756058/10471571237*a^8 + 2672491688/10471571237*a^7 - 1444137529/10471571237*a^6 + 1986766771/10471571237*a^5 + 3287951789/10471571237*a^4 - 1390306656/10471571237*a^3 + 282911940/10471571237*a^2 + 1424474522/10471571237*a - 2272174995/10471571237

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 $1$

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:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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:	$\frac{2763816314}{10471571237}a^{13}-\frac{10665468517}{10471571237}a^{12}-\frac{16668536112}{10471571237}a^{11}+\frac{69506535392}{10471571237}a^{10}+\frac{81188770045}{10471571237}a^{9}-\frac{120125850047}{10471571237}a^{8}-\frac{567319764631}{10471571237}a^{7}+\frac{507426018939}{10471571237}a^{6}+\frac{727111816258}{10471571237}a^{5}-\frac{528738567562}{10471571237}a^{4}-\frac{209176599586}{10471571237}a^{3}+\frac{65228535805}{10471571237}a^{2}-\frac{13248133878}{10471571237}a+\frac{11537552119}{10471571237}$, $\frac{6161012256}{10471571237}a^{13}-\frac{24975319031}{10471571237}a^{12}-\frac{33340055816}{10471571237}a^{11}+\frac{166083969277}{10471571237}a^{10}+\frac{154332709182}{10471571237}a^{9}-\frac{331743369994}{10471571237}a^{8}-\frac{1230860266466}{10471571237}a^{7}+\frac{1450421773658}{10471571237}a^{6}+\frac{1593250100809}{10471571237}a^{5}-\frac{1765525105961}{10471571237}a^{4}-\frac{543050899524}{10471571237}a^{3}+\frac{561168213051}{10471571237}a^{2}+\frac{56801266891}{10471571237}a-\frac{26445742605}{10471571237}$, $\frac{734710875}{10471571237}a^{13}+\frac{2125910506}{10471571237}a^{12}-\frac{23485727700}{10471571237}a^{11}-\frac{11157717658}{10471571237}a^{10}+\frac{144498358592}{10471571237}a^{9}+\frac{109921304817}{10471571237}a^{8}-\frac{359049870691}{10471571237}a^{7}-\frac{859303447620}{10471571237}a^{6}+\frac{1107964249976}{10471571237}a^{5}+\frac{1083207586678}{10471571237}a^{4}-\frac{1039279196259}{10471571237}a^{3}-\frac{293771566129}{10471571237}a^{2}+\frac{185609637475}{10471571237}a-\frac{11539282733}{10471571237}$, $\frac{1397116103}{10471571237}a^{13}-\frac{6334098659}{10471571237}a^{12}-\frac{4526318851}{10471571237}a^{11}+\frac{39933408642}{10471571237}a^{10}+\frac{15597823373}{10471571237}a^{9}-\frac{83162017514}{10471571237}a^{8}-\frac{237743573498}{10471571237}a^{7}+\frac{443325111604}{10471571237}a^{6}+\frac{145188348429}{10471571237}a^{5}-\frac{479316645330}{10471571237}a^{4}+\frac{132187852343}{10471571237}a^{3}+\frac{82410611334}{10471571237}a^{2}-\frac{52158795987}{10471571237}a+\frac{2888313997}{10471571237}$, $\frac{1919832779}{10471571237}a^{13}-\frac{6615547587}{10471571237}a^{12}-\frac{12993157833}{10471571237}a^{11}+\frac{38317633082}{10471571237}a^{10}+\frac{63195572579}{10471571237}a^{9}-\frac{28775012178}{10471571237}a^{8}-\frac{362816782957}{10471571237}a^{7}+\frac{165681807919}{10471571237}a^{6}+\frac{316012207736}{10471571237}a^{5}-\frac{67784281902}{10471571237}a^{4}+\frac{147547912661}{10471571237}a^{3}-\frac{81924585725}{10471571237}a^{2}-\frac{129948724008}{10471571237}a-\frac{854908902}{10471571237}$, $\frac{4005690627}{10471571237}a^{13}-\frac{16590650031}{10471571237}a^{12}-\frac{19497485179}{10471571237}a^{11}+\frac{107430923293}{10471571237}a^{10}+\frac{84722965331}{10471571237}a^{9}-\frac{208013481299}{10471571237}a^{8}-\frac{750289331904}{10471571237}a^{7}+\frac{990598132647}{10471571237}a^{6}+\frac{788565211984}{10471571237}a^{5}-\frac{1171793538554}{10471571237}a^{4}+\frac{20145302515}{10471571237}a^{3}+\frac{297629684887}{10471571237}a^{2}-\frac{110695482361}{10471571237}a+\frac{6687877957}{10471571237}$, $\frac{2763816314}{10471571237}a^{13}-\frac{10665468517}{10471571237}a^{12}-\frac{16668536112}{10471571237}a^{11}+\frac{69506535392}{10471571237}a^{10}+\frac{81188770045}{10471571237}a^{9}-\frac{120125850047}{10471571237}a^{8}-\frac{567319764631}{10471571237}a^{7}+\frac{507426018939}{10471571237}a^{6}+\frac{727111816258}{10471571237}a^{5}-\frac{528738567562}{10471571237}a^{4}-\frac{209176599586}{10471571237}a^{3}+\frac{65228535805}{10471571237}a^{2}-\frac{13248133878}{10471571237}a+\frac{22009123356}{10471571237}$, $\frac{5095166160}{10471571237}a^{13}-\frac{22267238092}{10471571237}a^{12}-\frac{22389363074}{10471571237}a^{11}+\frac{150487206877}{10471571237}a^{10}+\frac{94656508187}{10471571237}a^{9}-\frac{342338721054}{10471571237}a^{8}-\frac{983378561954}{10471571237}a^{7}+\frac{1551717814077}{10471571237}a^{6}+\frac{1217418887397}{10471571237}a^{5}-\frac{1987682036819}{10471571237}a^{4}-\frac{380026261334}{10471571237}a^{3}+\frac{711523910858}{10471571237}a^{2}+\frac{49151633899}{10471571237}a-\frac{56438211332}{10471571237}$, $\frac{6904280171}{10471571237}a^{13}-\frac{22346574132}{10471571237}a^{12}-\frac{56527515636}{10471571237}a^{11}+\frac{143014607631}{10471571237}a^{10}+\frac{296199907338}{10471571237}a^{9}-\frac{148303737140}{10471571237}a^{8}-\frac{1532385172064}{10471571237}a^{7}+\frac{392660387023}{10471571237}a^{6}+\frac{2288332211194}{10471571237}a^{5}-\frac{198435553028}{10471571237}a^{4}-\frac{985962269259}{10471571237}a^{3}-\frac{122403999566}{10471571237}a^{2}+\frac{43018353211}{10471571237}a+\frac{11947915797}{10471571237}$, $\frac{4160932417}{10471571237}a^{13}-\frac{16999567176}{10471571237}a^{12}-\frac{21194854963}{10471571237}a^{11}+\frac{109439944034}{10471571237}a^{10}+\frac{96786593418}{10471571237}a^{9}-\frac{203287867561}{10471571237}a^{8}-\frac{805063338129}{10471571237}a^{7}+\frac{950751130543}{10471571237}a^{6}+\frac{872300164687}{10471571237}a^{5}-\frac{1008055212892}{10471571237}a^{4}-\frac{76988747243}{10471571237}a^{3}+\frac{147639147139}{10471571237}a^{2}-\frac{75878501102}{10471571237}a+\frac{24897437353}{10471571237}$, $\frac{5426301381}{10471571237}a^{13}-\frac{27101229537}{10471571237}a^{12}-\frac{9854328116}{10471571237}a^{11}+\frac{177241686935}{10471571237}a^{10}+\frac{9834350590}{10471571237}a^{9}-\frac{441664674811}{10471571237}a^{8}-\frac{871810395775}{10471571237}a^{7}+\frac{2309725221278}{10471571237}a^{6}+\frac{485285850833}{10471571237}a^{5}-\frac{2848732692639}{10471571237}a^{4}+\frac{496228296735}{10471571237}a^{3}+\frac{854939779180}{10471571237}a^{2}-\frac{118336799347}{10471571237}a-\frac{14906459872}{10471571237}$ 2763816314/10471571237a^13 - 10665468517/10471571237a^12 - 16668536112/10471571237a^11 + 69506535392/10471571237a^10 + 81188770045/10471571237a^9 - 120125850047/10471571237a^8 - 567319764631/10471571237a^7 + 507426018939/10471571237a^6 + 727111816258/10471571237a^5 - 528738567562/10471571237a^4 - 209176599586/10471571237a^3 + 65228535805/10471571237a^2 - 13248133878/10471571237a + 11537552119/10471571237, 6161012256/10471571237a^13 - 24975319031/10471571237a^12 - 33340055816/10471571237a^11 + 166083969277/10471571237a^10 + 154332709182/10471571237a^9 - 331743369994/10471571237a^8 - 1230860266466/10471571237a^7 + 1450421773658/10471571237a^6 + 1593250100809/10471571237a^5 - 1765525105961/10471571237a^4 - 543050899524/10471571237a^3 + 561168213051/10471571237a^2 + 56801266891/10471571237a - 26445742605/10471571237, 734710875/10471571237a^13 + 2125910506/10471571237a^12 - 23485727700/10471571237a^11 - 11157717658/10471571237a^10 + 144498358592/10471571237a^9 + 109921304817/10471571237a^8 - 359049870691/10471571237a^7 - 859303447620/10471571237a^6 + 1107964249976/10471571237a^5 + 1083207586678/10471571237a^4 - 1039279196259/10471571237a^3 - 293771566129/10471571237a^2 + 185609637475/10471571237a - 11539282733/10471571237, 1397116103/10471571237a^13 - 6334098659/10471571237a^12 - 4526318851/10471571237a^11 + 39933408642/10471571237a^10 + 15597823373/10471571237a^9 - 83162017514/10471571237a^8 - 237743573498/10471571237a^7 + 443325111604/10471571237a^6 + 145188348429/10471571237a^5 - 479316645330/10471571237a^4 + 132187852343/10471571237a^3 + 82410611334/10471571237a^2 - 52158795987/10471571237a + 2888313997/10471571237, 1919832779/10471571237a^13 - 6615547587/10471571237a^12 - 12993157833/10471571237a^11 + 38317633082/10471571237a^10 + 63195572579/10471571237a^9 - 28775012178/10471571237a^8 - 362816782957/10471571237a^7 + 165681807919/10471571237a^6 + 316012207736/10471571237a^5 - 67784281902/10471571237a^4 + 147547912661/10471571237a^3 - 81924585725/10471571237a^2 - 129948724008/10471571237a - 854908902/10471571237, 4005690627/10471571237a^13 - 16590650031/10471571237a^12 - 19497485179/10471571237a^11 + 107430923293/10471571237a^10 + 84722965331/10471571237a^9 - 208013481299/10471571237a^8 - 750289331904/10471571237a^7 + 990598132647/10471571237a^6 + 788565211984/10471571237a^5 - 1171793538554/10471571237a^4 + 20145302515/10471571237a^3 + 297629684887/10471571237a^2 - 110695482361/10471571237a + 6687877957/10471571237, 2763816314/10471571237a^13 - 10665468517/10471571237a^12 - 16668536112/10471571237a^11 + 69506535392/10471571237a^10 + 81188770045/10471571237a^9 - 120125850047/10471571237a^8 - 567319764631/10471571237a^7 + 507426018939/10471571237a^6 + 727111816258/10471571237a^5 - 528738567562/10471571237a^4 - 209176599586/10471571237a^3 + 65228535805/10471571237a^2 - 13248133878/10471571237a + 22009123356/10471571237, 5095166160/10471571237a^13 - 22267238092/10471571237a^12 - 22389363074/10471571237a^11 + 150487206877/10471571237a^10 + 94656508187/10471571237a^9 - 342338721054/10471571237a^8 - 983378561954/10471571237a^7 + 1551717814077/10471571237a^6 + 1217418887397/10471571237a^5 - 1987682036819/10471571237a^4 - 380026261334/10471571237a^3 + 711523910858/10471571237a^2 + 49151633899/10471571237a - 56438211332/10471571237, 6904280171/10471571237a^13 - 22346574132/10471571237a^12 - 56527515636/10471571237a^11 + 143014607631/10471571237a^10 + 296199907338/10471571237a^9 - 148303737140/10471571237a^8 - 1532385172064/10471571237a^7 + 392660387023/10471571237a^6 + 2288332211194/10471571237a^5 - 198435553028/10471571237a^4 - 985962269259/10471571237a^3 - 122403999566/10471571237a^2 + 43018353211/10471571237a + 11947915797/10471571237, 4160932417/10471571237a^13 - 16999567176/10471571237a^12 - 21194854963/10471571237a^11 + 109439944034/10471571237a^10 + 96786593418/10471571237a^9 - 203287867561/10471571237a^8 - 805063338129/10471571237a^7 + 950751130543/10471571237a^6 + 872300164687/10471571237a^5 - 1008055212892/10471571237a^4 - 76988747243/10471571237a^3 + 147639147139/10471571237a^2 - 75878501102/10471571237a + 24897437353/10471571237, 5426301381/10471571237a^13 - 27101229537/10471571237a^12 - 9854328116/10471571237a^11 + 177241686935/10471571237a^10 + 9834350590/10471571237a^9 - 441664674811/10471571237a^8 - 871810395775/10471571237a^7 + 2309725221278/10471571237a^6 + 485285850833/10471571237a^5 - 2848732692639/10471571237a^4 + 496228296735/10471571237a^3 + 854939779180/10471571237a^2 - 118336799347/10471571237*a - 14906459872/10471571237	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:	$ 18052.2756731 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{2}\cdot 18052.2756731 \cdot 1}{2\cdot\sqrt{3161612016871142273}}\cr\approx \mathstrut & 0.205214180582 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 6*x^12 + 28*x^11 + 29*x^10 - 61*x^9 - 215*x^8 + 237*x^7 + 347*x^6 - 324*x^5 - 202*x^4 + 139*x^3 + 47*x^2 - 14*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^14 - 4*x^13 - 6*x^12 + 28*x^11 + 29*x^10 - 61*x^9 - 215*x^8 + 237*x^7 + 347*x^6 - 324*x^5 - 202*x^4 + 139*x^3 + 47*x^2 - 14*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^14 - 4*x^13 - 6*x^12 + 28*x^11 + 29*x^10 - 61*x^9 - 215*x^8 + 237*x^7 + 347*x^6 - 324*x^5 - 202*x^4 + 139*x^3 + 47*x^2 - 14*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^14 - 4*x^13 - 6*x^12 + 28*x^11 + 29*x^10 - 61*x^9 - 215*x^8 + 237*x^7 + 347*x^6 - 324*x^5 - 202*x^4 + 139*x^3 + 47*x^2 - 14*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

$C_2^7.S_7$ (as 14T57):

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 645120

The 110 conjugacy class representatives for $C_2^7.S_7$

Character table for $C_2^7.S_7$

Intermediate fields

7.7.25367689.1

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 14 sibling:	data not computed
Degree 28 siblings:	data not computed
Degree 42 siblings:	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	${\href{/padicField/2.7.0.1}{7} }^{2}$	${\href{/padicField/3.14.0.1}{14} }$	${\href{/padicField/5.14.0.1}{14} }$	${\href{/padicField/7.14.0.1}{14} }$	${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$	${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	R	${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.2.0.1}{2} }$	R	${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$	${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$	${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$	${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$	${\href{/padicField/43.7.0.1}{7} }^{2}$	${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$	${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$	${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$17$ 17	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.6.3.1	$x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$23$ 23	$\Q_{23}$	$x + 18$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{23}$	$x + 18$	$1$	$1$	$0$	Trivial	$[\ ]$
	23.2.0.1	$x^{2} + 21 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	23.2.1.1	$x^{2} + 115$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	23.2.1.1	$x^{2} + 115$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	23.3.0.1	$x^{3} + 2 x + 18$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	23.3.0.1	$x^{3} + 2 x + 18$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
$64879$ 64879	$\Q_{64879}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{64879}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $4$	$2$	$2$	$2$
		Deg $8$	$1$	$8$	$0$	$C_8$	$[\ ]^{8}$

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