# Properties

 Label 14.0.418...987.1 Degree $14$ Signature $[0, 7]$ Discriminant $-4.190\times 10^{23}$ Root discriminant $$48.67$$ Ramified primes $3,7$ Class number $203$ (GRH) Class group $[203]$ (GRH) Galois group $C_{14}$ (as 14T1)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409)

gp: K = bnfinit(y^14 + 21*y^12 - 42*y^11 + 350*y^10 - 553*y^9 + 2184*y^8 - 2696*y^7 + 8869*y^6 - 8701*y^5 + 18151*y^4 - 8246*y^3 + 17920*y^2 - 8148*y + 9409, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409)

$$x^{14} + 21 x^{12} - 42 x^{11} + 350 x^{10} - 553 x^{9} + 2184 x^{8} - 2696 x^{7} + 8869 x^{6} - 8701 x^{5} + 18151 x^{4} - 8246 x^{3} + 17920 x^{2} - 8148 x + 9409$$

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: $[0, 7]$ sage: K.signature()  gp: K.sign  magma: Signature(K);  oscar: signature(K) Discriminant: $$-418988153029298748294987$$ -418988153029298748294987 $$\medspace = -\,3^{7}\cdot 7^{24}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$48.67$$ 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)) Ramified primes: $$3$$, $$7$$ 3, 7 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{-3})$$ $\card{ \Gal(K/\Q) }$: $14$ sage: K.automorphisms()  magma: Automorphisms(K);  oscar: automorphisms(K) This field is Galois and abelian over $\Q$. Conductor: $$147=3\cdot 7^{2}$$ Dirichlet character group: $\lbrace$$\chi_{147}(64,·), \chi_{147}(1,·), \chi_{147}(134,·), \chi_{147}(71,·), \chi_{147}(8,·), \chi_{147}(106,·), \chi_{147}(43,·), \chi_{147}(113,·), \chi_{147}(50,·), \chi_{147}(85,·), \chi_{147}(22,·), \chi_{147}(92,·), \chi_{147}(29,·), \chi_{147}(127,·)$$\rbrace$ This is a CM field. Reflex fields: unavailable$^{64}$

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{589}a^{10}-\frac{268}{589}a^{9}-\frac{125}{589}a^{8}+\frac{265}{589}a^{7}-\frac{156}{589}a^{6}+\frac{23}{589}a^{5}-\frac{273}{589}a^{4}-\frac{198}{589}a^{3}+\frac{159}{589}a^{2}+\frac{26}{589}a-\frac{233}{589}$, $\frac{1}{589}a^{11}-\frac{91}{589}a^{9}-\frac{251}{589}a^{8}+\frac{184}{589}a^{7}+\frac{34}{589}a^{6}+\frac{1}{589}a^{5}+\frac{263}{589}a^{4}+\frac{105}{589}a^{3}+\frac{230}{589}a^{2}+\frac{256}{589}a-\frac{10}{589}$, $\frac{1}{589}a^{12}+\frac{99}{589}a^{9}-\frac{59}{589}a^{6}-\frac{118}{589}a^{3}+\frac{1}{589}$, $\frac{1}{18\!\cdots\!53}a^{13}+\frac{5828455076247}{19\!\cdots\!49}a^{12}-\frac{91476252389347}{18\!\cdots\!53}a^{11}-\frac{10\!\cdots\!66}{18\!\cdots\!53}a^{10}+\frac{77\!\cdots\!06}{18\!\cdots\!53}a^{9}+\frac{17\!\cdots\!44}{18\!\cdots\!53}a^{8}+\frac{97\!\cdots\!27}{18\!\cdots\!53}a^{7}-\frac{95\!\cdots\!11}{18\!\cdots\!53}a^{6}+\frac{29\!\cdots\!21}{18\!\cdots\!53}a^{5}+\frac{54\!\cdots\!43}{18\!\cdots\!53}a^{4}+\frac{61\!\cdots\!45}{18\!\cdots\!53}a^{3}-\frac{27\!\cdots\!95}{18\!\cdots\!53}a^{2}+\frac{49\!\cdots\!48}{18\!\cdots\!53}a-\frac{43\!\cdots\!41}{19\!\cdots\!49}$

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

$C_{203}$, which has order $203$ (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: $6$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K);  oscar: rank(UK) Torsion generator: $$-\frac{161289742466524}{1895673790255648453} a^{13} - \frac{957311663154}{19543028765522149} a^{12} - \frac{3252983796414282}{1895673790255648453} a^{11} + \frac{4785500909859803}{1895673790255648453} a^{10} - \frac{50010109733421474}{1895673790255648453} a^{9} + \frac{50386722190134414}{1895673790255648453} a^{8} - \frac{257564795016183278}{1895673790255648453} a^{7} + \frac{157348402783570740}{1895673790255648453} a^{6} - \frac{958061773957764798}{1895673790255648453} a^{5} + \frac{288377486595742334}{1895673790255648453} a^{4} - \frac{1285265248884962772}{1895673790255648453} a^{3} - \frac{1523096170019655210}{1895673790255648453} a^{2} - \frac{1081764845716455943}{1895673790255648453} a + \frac{2712612543346663}{19543028765522149}$$ -(161289742466524)/(1895673790255648453)*a^(13) - (957311663154)/(19543028765522149)*a^(12) - (3252983796414282)/(1895673790255648453)*a^(11) + (4785500909859803)/(1895673790255648453)*a^(10) - (50010109733421474)/(1895673790255648453)*a^(9) + (50386722190134414)/(1895673790255648453)*a^(8) - (257564795016183278)/(1895673790255648453)*a^(7) + (157348402783570740)/(1895673790255648453)*a^(6) - (958061773957764798)/(1895673790255648453)*a^(5) + (288377486595742334)/(1895673790255648453)*a^(4) - (1285265248884962772)/(1895673790255648453)*a^(3) - (1523096170019655210)/(1895673790255648453)*a^(2) - (1081764845716455943)/(1895673790255648453)*a + (2712612543346663)/(19543028765522149)  (order $6$) 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: $\frac{500894425233379}{18\!\cdots\!53}a^{13}+\frac{9083208803277}{19\!\cdots\!49}a^{12}+\frac{10\!\cdots\!04}{18\!\cdots\!53}a^{11}-\frac{48\!\cdots\!58}{18\!\cdots\!53}a^{10}+\frac{13\!\cdots\!29}{18\!\cdots\!53}a^{9}-\frac{30\!\cdots\!04}{18\!\cdots\!53}a^{8}+\frac{69\!\cdots\!92}{18\!\cdots\!53}a^{7}-\frac{14\!\cdots\!59}{18\!\cdots\!53}a^{6}+\frac{24\!\cdots\!59}{18\!\cdots\!53}a^{5}+\frac{14\!\cdots\!04}{18\!\cdots\!53}a^{4}+\frac{31\!\cdots\!67}{18\!\cdots\!53}a^{3}+\frac{41\!\cdots\!43}{18\!\cdots\!53}a^{2}+\frac{24\!\cdots\!42}{18\!\cdots\!53}a-\frac{16\!\cdots\!09}{19\!\cdots\!49}$, $\frac{1182799648905}{19\!\cdots\!49}a^{13}+\frac{8568865761786}{19\!\cdots\!49}a^{12}+\frac{7684286930910}{19\!\cdots\!49}a^{11}+\frac{77847407861811}{19\!\cdots\!49}a^{10}-\frac{326021747139292}{19\!\cdots\!49}a^{9}+\frac{21\!\cdots\!20}{19\!\cdots\!49}a^{8}-\frac{59\!\cdots\!51}{19\!\cdots\!49}a^{7}+\frac{10\!\cdots\!52}{19\!\cdots\!49}a^{6}-\frac{27\!\cdots\!81}{19\!\cdots\!49}a^{5}+\frac{43\!\cdots\!62}{19\!\cdots\!49}a^{4}-\frac{86\!\cdots\!40}{19\!\cdots\!49}a^{3}+\frac{47\!\cdots\!64}{19\!\cdots\!49}a^{2}-\frac{27\!\cdots\!47}{19\!\cdots\!49}a-\frac{14\!\cdots\!79}{19\!\cdots\!49}$, $\frac{339974479466749}{18\!\cdots\!53}a^{13}+\frac{5231873784525}{19\!\cdots\!49}a^{12}+\frac{64\!\cdots\!91}{18\!\cdots\!53}a^{11}-\frac{46\!\cdots\!05}{18\!\cdots\!53}a^{10}+\frac{87\!\cdots\!95}{18\!\cdots\!53}a^{9}+\frac{19\!\cdots\!79}{18\!\cdots\!53}a^{8}+\frac{35\!\cdots\!75}{18\!\cdots\!53}a^{7}+\frac{17\!\cdots\!34}{18\!\cdots\!53}a^{6}+\frac{16\!\cdots\!17}{18\!\cdots\!53}a^{5}+\frac{10\!\cdots\!11}{18\!\cdots\!53}a^{4}+\frac{24\!\cdots\!84}{18\!\cdots\!53}a^{3}+\frac{47\!\cdots\!73}{18\!\cdots\!53}a^{2}+\frac{49\!\cdots\!29}{18\!\cdots\!53}a+\frac{45\!\cdots\!05}{19\!\cdots\!49}$, $\frac{8195604655819}{19\!\cdots\!49}a^{13}+\frac{7724519215672}{19\!\cdots\!49}a^{12}+\frac{143285362101380}{19\!\cdots\!49}a^{11}-\frac{276452985460239}{19\!\cdots\!49}a^{10}+\frac{18\!\cdots\!69}{19\!\cdots\!49}a^{9}-\frac{23\!\cdots\!44}{19\!\cdots\!49}a^{8}+\frac{70\!\cdots\!03}{19\!\cdots\!49}a^{7}-\frac{13\!\cdots\!61}{19\!\cdots\!49}a^{6}+\frac{22\!\cdots\!81}{19\!\cdots\!49}a^{5}-\frac{27\!\cdots\!18}{19\!\cdots\!49}a^{4}+\frac{31\!\cdots\!55}{19\!\cdots\!49}a^{3}-\frac{30\!\cdots\!10}{19\!\cdots\!49}a^{2}+\frac{18\!\cdots\!25}{19\!\cdots\!49}a-\frac{28\!\cdots\!54}{19\!\cdots\!49}$, $\frac{18709735634464}{19\!\cdots\!49}a^{13}+\frac{5140286677554}{19\!\cdots\!49}a^{12}+\frac{344302418627400}{19\!\cdots\!49}a^{11}-\frac{786930198666752}{19\!\cdots\!49}a^{10}+\frac{52\!\cdots\!31}{19\!\cdots\!49}a^{9}-\frac{84\!\cdots\!64}{19\!\cdots\!49}a^{8}+\frac{25\!\cdots\!96}{19\!\cdots\!49}a^{7}-\frac{42\!\cdots\!12}{19\!\cdots\!49}a^{6}+\frac{92\!\cdots\!84}{19\!\cdots\!49}a^{5}-\frac{12\!\cdots\!64}{19\!\cdots\!49}a^{4}+\frac{98\!\cdots\!23}{19\!\cdots\!49}a^{3}-\frac{13\!\cdots\!32}{19\!\cdots\!49}a^{2}+\frac{82\!\cdots\!56}{19\!\cdots\!49}a-\frac{10\!\cdots\!06}{19\!\cdots\!49}$, $\frac{957311663154}{19\!\cdots\!49}a^{13}-\frac{1382482426626}{19\!\cdots\!49}a^{12}+\frac{20501734780765}{19\!\cdots\!49}a^{11}-\frac{66405155977958}{19\!\cdots\!49}a^{10}+\frac{400067065916014}{19\!\cdots\!49}a^{9}-\frac{976206211656754}{19\!\cdots\!49}a^{8}+\frac{28\!\cdots\!12}{19\!\cdots\!49}a^{7}-\frac{48\!\cdots\!14}{19\!\cdots\!49}a^{6}+\frac{11\!\cdots\!70}{19\!\cdots\!49}a^{5}-\frac{16\!\cdots\!16}{19\!\cdots\!49}a^{4}+\frac{29\!\cdots\!62}{19\!\cdots\!49}a^{3}-\frac{18\!\cdots\!21}{19\!\cdots\!49}a^{2}+\frac{10\!\cdots\!53}{19\!\cdots\!49}a+\frac{23\!\cdots\!70}{19\!\cdots\!49}$ 500894425233379/1895673790255648453*a^13 + 9083208803277/19543028765522149*a^12 + 10780000631860704/1895673790255648453*a^11 - 4814025083573158/1895673790255648453*a^10 + 138166532568995329/1895673790255648453*a^9 - 30725205647995604/1895673790255648453*a^8 + 691993708456829692/1895673790255648453*a^7 - 146804215598860759/1895673790255648453*a^6 + 2455814208030435859/1895673790255648453*a^5 + 14228154047680104/1895673790255648453*a^4 + 3141816885187212167/1895673790255648453*a^3 + 414893783043922843/1895673790255648453*a^2 + 2448568861516703842/1895673790255648453*a - 1688515671995309/19543028765522149, 1182799648905/19543028765522149*a^13 + 8568865761786/19543028765522149*a^12 + 7684286930910/19543028765522149*a^11 + 77847407861811/19543028765522149*a^10 - 326021747139292/19543028765522149*a^9 + 2105790296178120/19543028765522149*a^8 - 5999512245348951/19543028765522149*a^7 + 10658160586188552/19543028765522149*a^6 - 27201162035349981/19543028765522149*a^5 + 43336254027510162/19543028765522149*a^4 - 86422001002514940/19543028765522149*a^3 + 47887997336383764/19543028765522149*a^2 - 27003924383682147/19543028765522149*a - 14404017801319979/19543028765522149, 339974479466749/1895673790255648453*a^13 + 5231873784525/19543028765522149*a^12 + 6488036580977791/1895673790255648453*a^11 - 4678069123917005/1895673790255648453*a^10 + 87355820544429095/1895673790255648453*a^9 + 1909004606609079/1895673790255648453*a^8 + 351929344107517875/1895673790255648453*a^7 + 174724193320690034/1895673790255648453*a^6 + 1661793391752441517/1895673790255648453*a^5 + 1071267502399390111/1895673790255648453*a^4 + 2450993134440676084/1895673790255648453*a^3 + 4710204451097664973/1895673790255648453*a^2 + 4909241722594054029/1895673790255648453*a + 45571918943043605/19543028765522149, 8195604655819/19543028765522149*a^13 + 7724519215672/19543028765522149*a^12 + 143285362101380/19543028765522149*a^11 - 276452985460239/19543028765522149*a^10 + 1876682021950969/19543028765522149*a^9 - 2308121010496444/19543028765522149*a^8 + 7076803262843503/19543028765522149*a^7 - 13002562703347761/19543028765522149*a^6 + 22784924335655881/19543028765522149*a^5 - 27585183550455218/19543028765522149*a^4 + 3191695502824155/19543028765522149*a^3 - 30076075614077710/19543028765522149*a^2 + 18989564221741725/19543028765522149*a - 28959377050466654/19543028765522149, 18709735634464/19543028765522149*a^13 + 5140286677554/19543028765522149*a^12 + 344302418627400/19543028765522149*a^11 - 786930198666752/19543028765522149*a^10 + 5269077495044031/19543028765522149*a^9 - 8496676590811664/19543028765522149*a^8 + 25446517367648496/19543028765522149*a^7 - 42049417564510112/19543028765522149*a^6 + 92363345961799584/19543028765522149*a^5 - 125590671570759264/19543028765522149*a^4 + 98281627435257523/19543028765522149*a^3 - 137776304395533832/19543028765522149*a^2 + 82713057581340056/19543028765522149*a - 106138081125005406/19543028765522149, 957311663154/19543028765522149*a^13 - 1382482426626/19543028765522149*a^12 + 20501734780765/19543028765522149*a^11 - 66405155977958/19543028765522149*a^10 + 400067065916014/19543028765522149*a^9 - 976206211656754/19543028765522149*a^8 + 2860708689754412/19543028765522149*a^7 - 4870277855441614/19543028765522149*a^6 + 11494892397994670/19543028765522149*a^5 - 16930978006442416/19543028765522149*a^4 + 29413313261841362/19543028765522149*a^3 - 18644817930759321/19543028765522149*a^2 + 10835725823841353/19543028765522149*a + 23440952511791470/19543028765522149 (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: $$35256.6897369$$ (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)^{7}\cdot 35256.6897369 \cdot 203}{6\cdot\sqrt{418988153029298748294987}}\cr\approx \mathstrut & 0.712433820745 \end{aligned} (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409)

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 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409, 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 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409);

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 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409);

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

$C_{14}$ (as 14T1):

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 cyclic group of order 14 The 14 conjugacy class representatives for $C_{14}$ Character table for $C_{14}$

## Intermediate fields

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 ${\href{/padicField/2.14.0.1}{14} }$ R ${\href{/padicField/5.14.0.1}{14} }$ R ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.7.0.1}{7} }^{2}$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.1.0.1}{1} }^{14}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.14.0.1}{14} }$ ${\href{/padicField/31.1.0.1}{1} }^{14}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.14.0.1}{14} }$

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$LabelPolynomial$efc$Galois group Slope content $$3$$ 3.14.7.2$x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7} $$7$$ 7.7.12.1x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7$$[2]$7.7.12.1$x^{7} + 42 x^{6} + 7$$7$$1$$12$$C_7[2]$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$* 1.3.2t1.a.a$1 3 $$$\Q(\sqrt{-3})$$$C_2$(as 2T1)$1-1$* 1.49.7t1.a.a$1 7^{2}$7.7.13841287201.1$C_7$(as 7T1)$01$* 1.147.14t1.a.d$1 3 \cdot 7^{2}$14.0.418988153029298748294987.1$C_{14}$(as 14T1)$0-1$* 1.49.7t1.a.b$1 7^{2}$7.7.13841287201.1$C_7$(as 7T1)$01$* 1.147.14t1.a.f$1 3 \cdot 7^{2}$14.0.418988153029298748294987.1$C_{14}$(as 14T1)$0-1$* 1.49.7t1.a.c$1 7^{2}$7.7.13841287201.1$C_7$(as 7T1)$01$* 1.147.14t1.a.b$1 3 \cdot 7^{2}$14.0.418988153029298748294987.1$C_{14}$(as 14T1)$0-1$* 1.49.7t1.a.d$1 7^{2}$7.7.13841287201.1$C_7$(as 7T1)$01$* 1.147.14t1.a.c$1 3 \cdot 7^{2}$14.0.418988153029298748294987.1$C_{14}$(as 14T1)$0-1$* 1.49.7t1.a.e$1 7^{2}$7.7.13841287201.1$C_7$(as 7T1)$01$* 1.147.14t1.a.e$1 3 \cdot 7^{2}$14.0.418988153029298748294987.1$C_{14}$(as 14T1)$0-1$* 1.49.7t1.a.f$1 7^{2}$7.7.13841287201.1$C_7$(as 7T1)$01$* 1.147.14t1.a.a$1 3 \cdot 7^{2}$14.0.418988153029298748294987.1$C_{14}$(as 14T1)$0-1\$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.