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Properties

Label 22.0.439...467.1
Degree $22$
Signature $[0, 11]$
Discriminant $-4.395\times 10^{27}$
Root discriminant \(18.05\)
Ramified primes $3,19457,8095783$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times S_{11}$ (as 22T47)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 2*x^20 + x^19 + 2*x^18 - x^17 + 4*x^15 - 6*x^14 - 6*x^13 - 3*x^12 - x^11 - 2*x^10 - 8*x^9 + 8*x^8 + 5*x^7 + 7*x^6 + 8*x^5 + 8*x^4 + 6*x^3 + 4*x^2 + 2*x + 1)
 
gp: K = bnfinit(y^22 - y^21 + 2*y^20 + y^19 + 2*y^18 - y^17 + 4*y^15 - 6*y^14 - 6*y^13 - 3*y^12 - y^11 - 2*y^10 - 8*y^9 + 8*y^8 + 5*y^7 + 7*y^6 + 8*y^5 + 8*y^4 + 6*y^3 + 4*y^2 + 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 + 2*x^20 + x^19 + 2*x^18 - x^17 + 4*x^15 - 6*x^14 - 6*x^13 - 3*x^12 - x^11 - 2*x^10 - 8*x^9 + 8*x^8 + 5*x^7 + 7*x^6 + 8*x^5 + 8*x^4 + 6*x^3 + 4*x^2 + 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 + 2*x^20 + x^19 + 2*x^18 - x^17 + 4*x^15 - 6*x^14 - 6*x^13 - 3*x^12 - x^11 - 2*x^10 - 8*x^9 + 8*x^8 + 5*x^7 + 7*x^6 + 8*x^5 + 8*x^4 + 6*x^3 + 4*x^2 + 2*x + 1)
 

\( x^{22} - x^{21} + 2 x^{20} + x^{19} + 2 x^{18} - x^{17} + 4 x^{15} - 6 x^{14} - 6 x^{13} - 3 x^{12} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-4395449323362095410329595467\) \(\medspace = -\,3^{11}\cdot 19457^{2}\cdot 8095783^{2}\) 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:  \(18.05\)
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:  $3^{1/2}19457^{1/2}8095783^{1/2}\approx 687429.2323526837$
Ramified primes:   \(3\), \(19457\), \(8095783\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3207649}a^{21}+\frac{1593331}{3207649}a^{20}-\frac{1362752}{3207649}a^{19}-\frac{1023881}{3207649}a^{18}-\frac{948931}{3207649}a^{17}-\frac{1487804}{3207649}a^{16}-\frac{844213}{3207649}a^{15}-\frac{17807}{3207649}a^{14}-\frac{807525}{3207649}a^{13}-\frac{48777}{3207649}a^{12}+\frac{172654}{3207649}a^{11}+\frac{749589}{3207649}a^{10}-\frac{1511061}{3207649}a^{9}+\frac{1002352}{3207649}a^{8}+\frac{30553}{139463}a^{7}-\frac{497876}{3207649}a^{6}-\frac{1296284}{3207649}a^{5}+\frac{828118}{3207649}a^{4}+\frac{493034}{3207649}a^{3}+\frac{778598}{3207649}a^{2}+\frac{442492}{3207649}a-\frac{1379205}{3207649}$ 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:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{2168402}{3207649} a^{21} + \frac{2272030}{3207649} a^{20} - \frac{4364211}{3207649} a^{19} - \frac{1477784}{3207649} a^{18} - \frac{4463450}{3207649} a^{17} + \frac{3242127}{3207649} a^{16} + \frac{703922}{3207649} a^{15} - \frac{7359546}{3207649} a^{14} + \frac{12104791}{3207649} a^{13} + \frac{11956824}{3207649} a^{12} + \frac{7097074}{3207649} a^{11} - \frac{1516657}{3207649} a^{10} - \frac{97786}{3207649} a^{9} + \frac{13711492}{3207649} a^{8} - \frac{682786}{139463} a^{7} - \frac{9525076}{3207649} a^{6} - \frac{17246426}{3207649} a^{5} - \frac{12325448}{3207649} a^{4} - \frac{12161160}{3207649} a^{3} - \frac{5901034}{3207649} a^{2} - \frac{6115361}{3207649} a + \frac{89366}{3207649} \)  (order $6$) 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:   $\frac{89366}{3207649}a^{21}+\frac{2079036}{3207649}a^{20}-\frac{2093298}{3207649}a^{19}+\frac{4453577}{3207649}a^{18}+\frac{1656516}{3207649}a^{17}+\frac{4374084}{3207649}a^{16}-\frac{3242127}{3207649}a^{15}-\frac{346458}{3207649}a^{14}+\frac{6823350}{3207649}a^{13}-\frac{12640987}{3207649}a^{12}-\frac{12224922}{3207649}a^{11}-\frac{7186440}{3207649}a^{10}+\frac{1337925}{3207649}a^{9}-\frac{617142}{3207649}a^{8}-\frac{565068}{139463}a^{7}+\frac{16150908}{3207649}a^{6}+\frac{10150638}{3207649}a^{5}+\frac{17961354}{3207649}a^{4}+\frac{13040376}{3207649}a^{3}+\frac{12697356}{3207649}a^{2}+\frac{6258498}{3207649}a+\frac{6294093}{3207649}$, $\frac{1133880}{3207649}a^{21}-\frac{1199639}{3207649}a^{20}+\frac{1061467}{3207649}a^{19}+\frac{2252535}{3207649}a^{18}-\frac{101720}{3207649}a^{17}-\frac{1983897}{3207649}a^{16}-\frac{3206562}{3207649}a^{15}+\frac{7564593}{3207649}a^{14}-\frac{6624652}{3207649}a^{13}-\frac{10603649}{3207649}a^{12}+\frac{2891401}{3207649}a^{11}+\frac{6804492}{3207649}a^{10}+\frac{3866670}{3207649}a^{9}-\frac{12969112}{3207649}a^{8}+\frac{408051}{139463}a^{7}+\frac{14585120}{3207649}a^{6}-\frac{2331246}{3207649}a^{5}+\frac{1723123}{3207649}a^{4}-\frac{3714045}{3207649}a^{3}+\frac{1881268}{3207649}a^{2}-\frac{4419971}{3207649}a-\frac{2187238}{3207649}$, $\frac{657061}{3207649}a^{21}-\frac{28078}{3207649}a^{20}+\frac{818829}{3207649}a^{19}+\frac{781625}{3207649}a^{18}+\frac{3676127}{3207649}a^{17}-\frac{2044208}{3207649}a^{16}-\frac{696423}{3207649}a^{15}+\frac{1218325}{3207649}a^{14}+\frac{75310}{3207649}a^{13}-\frac{8258536}{3207649}a^{12}-\frac{10335236}{3207649}a^{11}+\frac{7232224}{3207649}a^{10}+\frac{1135600}{3207649}a^{9}-\frac{7331102}{3207649}a^{8}+\frac{43735}{139463}a^{7}+\frac{10011425}{3207649}a^{6}+\frac{15038838}{3207649}a^{5}-\frac{5496917}{3207649}a^{4}+\frac{9732915}{3207649}a^{3}+\frac{4856766}{3207649}a^{2}-\frac{276997}{3207649}a-\frac{28674}{3207649}$, $\frac{18040}{3207649}a^{21}-\frac{51449}{3207649}a^{20}-\frac{624144}{3207649}a^{19}+\frac{2037351}{3207649}a^{18}-\frac{2700176}{3207649}a^{17}+\frac{1622672}{3207649}a^{16}+\frac{314932}{3207649}a^{15}+\frac{2734269}{3207649}a^{14}-\frac{1816891}{3207649}a^{13}-\frac{4248903}{3207649}a^{12}+\frac{9673928}{3207649}a^{11}-\frac{4070273}{3207649}a^{10}-\frac{7354536}{3207649}a^{9}-\frac{2294982}{3207649}a^{8}+\frac{157807}{139463}a^{7}+\frac{6149458}{3207649}a^{6}-\frac{14032746}{3207649}a^{5}+\frac{7642625}{3207649}a^{4}+\frac{5937981}{3207649}a^{3}+\frac{2820598}{3207649}a^{2}+\frac{1924968}{3207649}a+\frac{875093}{3207649}$, $\frac{66774}{3207649}a^{21}+\frac{1782162}{3207649}a^{20}-\frac{1815216}{3207649}a^{19}+\frac{2408541}{3207649}a^{18}+\frac{3187401}{3207649}a^{17}+\frac{680532}{3207649}a^{16}-\frac{3462985}{3207649}a^{15}-\frac{2214488}{3207649}a^{14}+\frac{8528287}{3207649}a^{13}-\frac{10894610}{3207649}a^{12}-\frac{15530555}{3207649}a^{11}+\frac{4108539}{3207649}a^{10}+\frac{6635028}{3207649}a^{9}+\frac{248414}{3207649}a^{8}-\frac{616057}{139463}a^{7}+\frac{14940457}{3207649}a^{6}+\frac{16378694}{3207649}a^{5}+\frac{90221}{3207649}a^{4}+\frac{8165927}{3207649}a^{3}+\frac{3735509}{3207649}a^{2}-\frac{1901780}{3207649}a-\frac{224231}{3207649}$, $\frac{7674}{139463}a^{21}-\frac{56968}{139463}a^{20}+\frac{153133}{139463}a^{19}-\frac{196300}{139463}a^{18}+\frac{103514}{139463}a^{17}+\frac{9525}{139463}a^{16}+\frac{123640}{139463}a^{15}-\frac{116641}{139463}a^{14}-\frac{326834}{139463}a^{13}+\frac{701309}{139463}a^{12}-\frac{230630}{139463}a^{11}-\frac{502764}{139463}a^{10}+\frac{47947}{139463}a^{9}+\frac{246409}{139463}a^{8}+\frac{468174}{139463}a^{7}-\frac{1087780}{139463}a^{6}+\frac{491300}{139463}a^{5}+\frac{345937}{139463}a^{4}-\frac{88274}{139463}a^{3}+\frac{87206}{139463}a^{2}+\frac{38484}{139463}a-\frac{32637}{139463}$, $\frac{2984233}{3207649}a^{21}-\frac{3583272}{3207649}a^{20}+\frac{6595997}{3207649}a^{19}+\frac{1116710}{3207649}a^{18}+\frac{6430588}{3207649}a^{17}-\frac{6239757}{3207649}a^{16}+\frac{930408}{3207649}a^{15}+\frac{10506899}{3207649}a^{14}-\frac{19458499}{3207649}a^{13}-\frac{14859666}{3207649}a^{12}-\frac{8102137}{3207649}a^{11}+\frac{4385915}{3207649}a^{10}-\frac{3437576}{3207649}a^{9}-\frac{21913040}{3207649}a^{8}+\frac{1242654}{139463}a^{7}+\frac{11442990}{3207649}a^{6}+\frac{24034424}{3207649}a^{5}+\frac{12006179}{3207649}a^{4}+\frac{18228410}{3207649}a^{3}+\frac{9177449}{3207649}a^{2}+\frac{6364806}{3207649}a+\frac{78393}{3207649}$, $\frac{1133880}{3207649}a^{21}-\frac{1199639}{3207649}a^{20}+\frac{1061467}{3207649}a^{19}+\frac{2252535}{3207649}a^{18}-\frac{101720}{3207649}a^{17}-\frac{1983897}{3207649}a^{16}-\frac{3206562}{3207649}a^{15}+\frac{7564593}{3207649}a^{14}-\frac{6624652}{3207649}a^{13}-\frac{10603649}{3207649}a^{12}+\frac{2891401}{3207649}a^{11}+\frac{6804492}{3207649}a^{10}+\frac{3866670}{3207649}a^{9}-\frac{12969112}{3207649}a^{8}+\frac{408051}{139463}a^{7}+\frac{14585120}{3207649}a^{6}-\frac{2331246}{3207649}a^{5}+\frac{1723123}{3207649}a^{4}-\frac{3714045}{3207649}a^{3}+\frac{1881268}{3207649}a^{2}-\frac{4419971}{3207649}a+\frac{1020411}{3207649}$, $\frac{1102728}{3207649}a^{21}-\frac{1486325}{3207649}a^{20}+\frac{3484905}{3207649}a^{19}-\frac{1875858}{3207649}a^{18}+\frac{5718906}{3207649}a^{17}-\frac{4441739}{3207649}a^{16}+\frac{2617961}{3207649}a^{15}+\frac{949682}{3207649}a^{14}-\frac{4989310}{3207649}a^{13}-\frac{1905224}{3207649}a^{12}-\frac{12436879}{3207649}a^{11}+\frac{4085035}{3207649}a^{10}-\frac{2225431}{3207649}a^{9}-\frac{8567952}{3207649}a^{8}+\frac{316507}{139463}a^{7}-\frac{3810537}{3207649}a^{6}+\frac{24075853}{3207649}a^{5}-\frac{3103204}{3207649}a^{4}+\frac{14760093}{3207649}a^{3}+\frac{3238110}{3207649}a^{2}+\frac{3959945}{3207649}a-\frac{443784}{3207649}$, $\frac{10504258}{3207649}a^{21}-\frac{18026981}{3207649}a^{20}+\frac{31687198}{3207649}a^{19}-\frac{7432503}{3207649}a^{18}+\frac{17372282}{3207649}a^{17}-\frac{17726367}{3207649}a^{16}+\frac{6397605}{3207649}a^{15}+\frac{43221017}{3207649}a^{14}-\frac{99018764}{3207649}a^{13}+\frac{1205251}{3207649}a^{12}-\frac{8999166}{3207649}a^{11}-\frac{13526265}{3207649}a^{10}-\frac{4668588}{3207649}a^{9}-\frac{76467055}{3207649}a^{8}+\frac{6152093}{139463}a^{7}-\frac{34949918}{3207649}a^{6}+\frac{64657198}{3207649}a^{5}+\frac{60486357}{3207649}a^{4}+\frac{17776981}{3207649}a^{3}+\frac{41992037}{3207649}a^{2}+\frac{6347486}{3207649}a+\frac{10804901}{3207649}$ 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:  \( 136255.877931 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 136255.877931 \cdot 1}{6\cdot\sqrt{4395449323362095410329595467}}\cr\approx \mathstrut & 0.206386320898 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 2*x^20 + x^19 + 2*x^18 - x^17 + 4*x^15 - 6*x^14 - 6*x^13 - 3*x^12 - x^11 - 2*x^10 - 8*x^9 + 8*x^8 + 5*x^7 + 7*x^6 + 8*x^5 + 8*x^4 + 6*x^3 + 4*x^2 + 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^22 - x^21 + 2*x^20 + x^19 + 2*x^18 - x^17 + 4*x^15 - 6*x^14 - 6*x^13 - 3*x^12 - x^11 - 2*x^10 - 8*x^9 + 8*x^8 + 5*x^7 + 7*x^6 + 8*x^5 + 8*x^4 + 6*x^3 + 4*x^2 + 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^22 - x^21 + 2*x^20 + x^19 + 2*x^18 - x^17 + 4*x^15 - 6*x^14 - 6*x^13 - 3*x^12 - x^11 - 2*x^10 - 8*x^9 + 8*x^8 + 5*x^7 + 7*x^6 + 8*x^5 + 8*x^4 + 6*x^3 + 4*x^2 + 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^22 - x^21 + 2*x^20 + x^19 + 2*x^18 - x^17 + 4*x^15 - 6*x^14 - 6*x^13 - 3*x^12 - x^11 - 2*x^10 - 8*x^9 + 8*x^8 + 5*x^7 + 7*x^6 + 8*x^5 + 8*x^4 + 6*x^3 + 4*x^2 + 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

$C_2\times S_{11}$ (as 22T47):

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 79833600
The 112 conjugacy class representatives for $C_2\times S_{11}$
Character table for $C_2\times S_{11}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.5.157519649831.1

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 22 sibling: data not computed
Degree 44 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 $22$ R $22$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ $22$ ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ $22$ ${\href{/padicField/43.9.0.1}{9} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ $18{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ $18{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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
\(3\) Copy content Toggle raw display 3.22.11.2$x^{22} + 33 x^{20} + 495 x^{18} + 4455 x^{16} + 26730 x^{14} + 4 x^{13} + 112266 x^{12} - 406 x^{11} + 336798 x^{10} + 1650 x^{9} + 721710 x^{8} + 20196 x^{7} + 1082565 x^{6} - 35640 x^{5} + 1082569 x^{4} - 37422 x^{3} + 649567 x^{2} + 20898 x + 177172$$2$$11$$11$22T1$[\ ]_{2}^{11}$
\(19457\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
\(8095783\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$