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Properties

Label 28.28.891...125.1
Degree $28$
Signature $[28, 0]$
Discriminant $8.913\times 10^{55}$
Root discriminant \(99.59\)
Ramified primes $3,5,29$
Class number not computed
Class group not computed
Galois group $C_{28}$ (as 28T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 115*x^26 + 115*x^25 + 5917*x^24 - 5917*x^23 - 179683*x^22 + 179683*x^21 + 3576861*x^20 - 3576861*x^19 - 49014755*x^18 + 49014755*x^17 + 472328221*x^16 - 472328221*x^15 - 3210926051*x^14 + 3210926051*x^13 + 15205345309*x^12 - 15205345309*x^11 - 48637728739*x^10 + 48637728739*x^9 + 99209390109*x^8 - 99209390109*x^7 - 115840964579*x^6 + 115840964579*x^5 + 61259327517*x^4 - 61259327517*x^3 - 6856169443*x^2 + 6856169443*x + 928458781)
 
gp: K = bnfinit(y^28 - y^27 - 115*y^26 + 115*y^25 + 5917*y^24 - 5917*y^23 - 179683*y^22 + 179683*y^21 + 3576861*y^20 - 3576861*y^19 - 49014755*y^18 + 49014755*y^17 + 472328221*y^16 - 472328221*y^15 - 3210926051*y^14 + 3210926051*y^13 + 15205345309*y^12 - 15205345309*y^11 - 48637728739*y^10 + 48637728739*y^9 + 99209390109*y^8 - 99209390109*y^7 - 115840964579*y^6 + 115840964579*y^5 + 61259327517*y^4 - 61259327517*y^3 - 6856169443*y^2 + 6856169443*y + 928458781, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - x^27 - 115*x^26 + 115*x^25 + 5917*x^24 - 5917*x^23 - 179683*x^22 + 179683*x^21 + 3576861*x^20 - 3576861*x^19 - 49014755*x^18 + 49014755*x^17 + 472328221*x^16 - 472328221*x^15 - 3210926051*x^14 + 3210926051*x^13 + 15205345309*x^12 - 15205345309*x^11 - 48637728739*x^10 + 48637728739*x^9 + 99209390109*x^8 - 99209390109*x^7 - 115840964579*x^6 + 115840964579*x^5 + 61259327517*x^4 - 61259327517*x^3 - 6856169443*x^2 + 6856169443*x + 928458781);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - x^27 - 115*x^26 + 115*x^25 + 5917*x^24 - 5917*x^23 - 179683*x^22 + 179683*x^21 + 3576861*x^20 - 3576861*x^19 - 49014755*x^18 + 49014755*x^17 + 472328221*x^16 - 472328221*x^15 - 3210926051*x^14 + 3210926051*x^13 + 15205345309*x^12 - 15205345309*x^11 - 48637728739*x^10 + 48637728739*x^9 + 99209390109*x^8 - 99209390109*x^7 - 115840964579*x^6 + 115840964579*x^5 + 61259327517*x^4 - 61259327517*x^3 - 6856169443*x^2 + 6856169443*x + 928458781)
 

\( x^{28} - x^{27} - 115 x^{26} + 115 x^{25} + 5917 x^{24} - 5917 x^{23} - 179683 x^{22} + 179683 x^{21} + \cdots + 928458781 \) Copy content Toggle raw display

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[28, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(89129930946082115284480820405277744589647154669189453125\) \(\medspace = 3^{14}\cdot 5^{14}\cdot 29^{27}\) 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:  \(99.59\)
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:  $3^{1/2}5^{1/2}29^{27/28}\approx 99.58986129486638$
Ramified primes:   \(3\), \(5\), \(29\) 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:  \(\Q(\sqrt{29}) \)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(196,·)$, $\chi_{435}(389,·)$, $\chi_{435}(134,·)$, $\chi_{435}(136,·)$, $\chi_{435}(329,·)$, $\chi_{435}(269,·)$, $\chi_{435}(14,·)$, $\chi_{435}(16,·)$, $\chi_{435}(404,·)$, $\chi_{435}(151,·)$, $\chi_{435}(89,·)$, $\chi_{435}(91,·)$, $\chi_{435}(286,·)$, $\chi_{435}(224,·)$, $\chi_{435}(226,·)$, $\chi_{435}(164,·)$, $\chi_{435}(359,·)$, $\chi_{435}(104,·)$, $\chi_{435}(361,·)$, $\chi_{435}(44,·)$, $\chi_{435}(241,·)$, $\chi_{435}(181,·)$, $\chi_{435}(374,·)$, $\chi_{435}(119,·)$, $\chi_{435}(376,·)$, $\chi_{435}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{139256549}a^{15}+\frac{9004948}{139256549}a^{14}-\frac{60}{139256549}a^{13}+\frac{52749108}{139256549}a^{12}+\frac{1440}{139256549}a^{11}-\frac{46427984}{139256549}a^{10}-\frac{17600}{139256549}a^{9}-\frac{12560039}{139256549}a^{8}+\frac{115200}{139256549}a^{7}-\frac{13217711}{139256549}a^{6}-\frac{387072}{139256549}a^{5}-\frac{57590470}{139256549}a^{4}+\frac{573440}{139256549}a^{3}+\frac{57590470}{139256549}a^{2}-\frac{245760}{139256549}a+\frac{10491267}{139256549}$, $\frac{1}{139256549}a^{16}-\frac{64}{139256549}a^{14}+\frac{36019792}{139256549}a^{13}+\frac{1664}{139256549}a^{12}-\frac{62694047}{139256549}a^{11}-\frac{22528}{139256549}a^{10}+\frac{571999}{139256549}a^{9}+\frac{168960}{139256549}a^{8}-\frac{61193810}{139256549}a^{7}-\frac{688128}{139256549}a^{6}+\frac{53476865}{139256549}a^{5}+\frac{1376256}{139256549}a^{4}+\frac{32302819}{139256549}a^{3}-\frac{1048576}{139256549}a^{2}+\frac{1435039}{139256549}a+\frac{131072}{139256549}$, $\frac{1}{139256549}a^{17}+\frac{55310268}{139256549}a^{14}-\frac{2176}{139256549}a^{13}-\frac{28908311}{139256549}a^{12}+\frac{69632}{139256549}a^{11}-\frac{46431448}{139256549}a^{10}-\frac{957440}{139256549}a^{9}-\frac{29497012}{139256549}a^{8}+\frac{6684672}{139256549}a^{7}+\frac{43082655}{139256549}a^{6}-\frac{23396352}{139256549}a^{5}-\frac{32816987}{139256549}a^{4}+\frac{35651584}{139256549}a^{3}+\frac{66554845}{139256549}a^{2}-\frac{15597568}{139256549}a-\frac{24841657}{139256549}$, $\frac{1}{139256549}a^{18}-\frac{2448}{139256549}a^{14}-\frac{52449407}{139256549}a^{13}+\frac{84864}{139256549}a^{12}-\frac{38471340}{139256549}a^{11}-\frac{1292544}{139256549}a^{10}+\frac{27942278}{139256549}a^{9}+\frac{10340352}{139256549}a^{8}-\frac{16391450}{139256549}a^{7}-\frac{43868160}{139256549}a^{6}-\frac{91853}{139256549}a^{5}-\frac{49013477}{139256549}a^{4}+\frac{18073165}{139256549}a^{3}+\frac{69067493}{139256549}a^{2}+\frac{55617584}{139256549}a+\frac{8912896}{139256549}$, $\frac{1}{139256549}a^{19}-\frac{10871445}{139256549}a^{14}-\frac{62016}{139256549}a^{13}+\frac{524121}{139256549}a^{12}+\frac{2232576}{139256549}a^{11}+\frac{5581430}{139256549}a^{10}-\frac{32744448}{139256549}a^{9}+\frac{12330407}{139256549}a^{8}-\frac{40371658}{139256549}a^{7}-\frac{49529013}{139256549}a^{6}-\frac{21769890}{139256549}a^{5}-\frac{35769807}{139256549}a^{4}-\frac{58973426}{139256549}a^{3}-\frac{29795993}{139256549}a^{2}-\frac{35681388}{139256549}a+\frac{59416600}{139256549}$, $\frac{1}{139256549}a^{20}-\frac{72960}{139256549}a^{14}+\frac{44520166}{139256549}a^{13}+\frac{2845440}{139256549}a^{12}+\frac{63728742}{139256549}a^{11}-\frac{46227456}{139256549}a^{10}+\frac{13396733}{139256549}a^{9}-\frac{32540847}{139256549}a^{8}+\frac{6789830}{139256549}a^{7}-\frac{9919812}{139256549}a^{6}-\frac{13331165}{139256549}a^{5}+\frac{48682915}{139256549}a^{4}-\frac{6304276}{139256549}a^{3}-\frac{4081180}{139256549}a^{2}+\frac{69242514}{139256549}a-\frac{59156655}{139256549}$, $\frac{1}{139256549}a^{21}+\frac{33128064}{139256549}a^{14}-\frac{1532160}{139256549}a^{13}+\frac{5403709}{139256549}a^{12}+\frac{58834944}{139256549}a^{11}+\frac{43238518}{139256549}a^{10}-\frac{63327906}{139256549}a^{9}-\frac{65563190}{139256549}a^{8}+\frac{39679248}{139256549}a^{7}-\frac{25923900}{139256549}a^{6}-\frac{62267307}{139256549}a^{5}-\frac{19142499}{139256549}a^{4}+\frac{57136520}{139256549}a^{3}-\frac{57175812}{139256549}a^{2}-\frac{25711434}{139256549}a-\frac{50409533}{139256549}$, $\frac{1}{139256549}a^{22}-\frac{1872640}{139256549}a^{14}+\frac{43495863}{139256549}a^{13}-\frac{61354725}{139256549}a^{12}-\frac{35433884}{139256549}a^{11}-\frac{65029619}{139256549}a^{10}+\frac{60449096}{139256549}a^{9}+\frac{20264331}{139256549}a^{8}-\frac{53171355}{139256549}a^{7}-\frac{64949011}{139256549}a^{6}+\frac{44557640}{139256549}a^{5}+\frac{17868881}{139256549}a^{4}-\frac{53551039}{139256549}a^{3}+\frac{13556205}{139256549}a^{2}+\frac{7718371}{139256549}a+\frac{16324720}{139256549}$, $\frac{1}{139256549}a^{23}-\frac{63226023}{139256549}a^{14}-\frac{34456576}{139256549}a^{13}-\frac{47039875}{139256549}a^{12}-\frac{14302450}{139256549}a^{11}+\frac{37267800}{139256549}a^{10}+\frac{65602444}{139256549}a^{9}-\frac{53478215}{139256549}a^{8}-\frac{45215412}{139256549}a^{7}+\frac{46276056}{139256549}a^{6}+\frac{1696346}{139256549}a^{5}-\frac{11714632}{139256549}a^{4}+\frac{52988466}{139256549}a^{3}-\frac{34118036}{139256549}a^{2}+\frac{39212765}{139256549}a+\frac{52301960}{139256549}$, $\frac{1}{139256549}a^{24}-\frac{43524096}{139256549}a^{14}+\frac{58582117}{139256549}a^{13}-\frac{63547526}{139256549}a^{12}+\frac{8957874}{139256549}a^{11}+\frac{34941724}{139256549}a^{10}-\frac{32399956}{139256549}a^{9}-\frac{27226987}{139256549}a^{8}+\frac{9586760}{139256549}a^{7}+\frac{40217715}{139256549}a^{6}+\frac{50288521}{139256549}a^{5}-\frac{3316335}{139256549}a^{4}+\frac{18439640}{139256549}a^{3}-\frac{43738983}{139256549}a^{2}+\frac{9883449}{139256549}a-\frac{48980206}{139256549}$, $\frac{1}{139256549}a^{25}+\frac{14397487}{139256549}a^{14}-\frac{29118855}{139256549}a^{13}-\frac{23320728}{139256549}a^{12}+\frac{44192914}{139256549}a^{11}-\frac{18553457}{139256549}a^{10}-\frac{1040538}{139256549}a^{9}+\frac{61089024}{139256549}a^{8}-\frac{55226379}{139256549}a^{7}-\frac{15313679}{139256549}a^{6}+\frac{16581675}{139256549}a^{5}+\frac{54122291}{139256549}a^{4}+\frac{19620183}{139256549}a^{3}-\frac{25799202}{139256549}a^{2}+\frac{43228622}{139256549}a-\frac{8384113}{139256549}$, $\frac{1}{139256549}a^{26}+\frac{31773763}{139256549}a^{14}+\frac{4989198}{139256549}a^{13}-\frac{23636518}{139256549}a^{12}-\frac{1708936}{139256549}a^{11}-\frac{40615073}{139256549}a^{10}+\frac{9941044}{139256549}a^{9}-\frac{41273826}{139256549}a^{8}-\frac{60317489}{139256549}a^{7}-\frac{33701312}{139256549}a^{6}+\frac{10375924}{139256549}a^{5}+\frac{60870449}{139256549}a^{4}-\frac{17723919}{139256549}a^{3}+\frac{1978356}{139256549}a^{2}-\frac{51632534}{139256549}a-\frac{61472552}{139256549}$, $\frac{1}{139256549}a^{27}+\frac{22451391}{139256549}a^{14}-\frac{66802424}{139256549}a^{13}-\frac{4795901}{139256549}a^{12}+\frac{20570828}{139256549}a^{11}-\frac{61402275}{139256549}a^{10}+\frac{61910739}{139256549}a^{9}+\frac{34350205}{139256549}a^{8}-\frac{12808447}{139256549}a^{7}-\frac{18439586}{139256549}a^{6}-\frac{65032197}{139256549}a^{5}+\frac{59117441}{139256549}a^{4}-\frac{17805204}{139256549}a^{3}+\frac{10782655}{139256549}a^{2}-\frac{13206298}{139256549}a+\frac{4019617}{139256549}$ 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

not computed

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:  $27$
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 115*x^26 + 115*x^25 + 5917*x^24 - 5917*x^23 - 179683*x^22 + 179683*x^21 + 3576861*x^20 - 3576861*x^19 - 49014755*x^18 + 49014755*x^17 + 472328221*x^16 - 472328221*x^15 - 3210926051*x^14 + 3210926051*x^13 + 15205345309*x^12 - 15205345309*x^11 - 48637728739*x^10 + 48637728739*x^9 + 99209390109*x^8 - 99209390109*x^7 - 115840964579*x^6 + 115840964579*x^5 + 61259327517*x^4 - 61259327517*x^3 - 6856169443*x^2 + 6856169443*x + 928458781)
 
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^28 - x^27 - 115*x^26 + 115*x^25 + 5917*x^24 - 5917*x^23 - 179683*x^22 + 179683*x^21 + 3576861*x^20 - 3576861*x^19 - 49014755*x^18 + 49014755*x^17 + 472328221*x^16 - 472328221*x^15 - 3210926051*x^14 + 3210926051*x^13 + 15205345309*x^12 - 15205345309*x^11 - 48637728739*x^10 + 48637728739*x^9 + 99209390109*x^8 - 99209390109*x^7 - 115840964579*x^6 + 115840964579*x^5 + 61259327517*x^4 - 61259327517*x^3 - 6856169443*x^2 + 6856169443*x + 928458781, 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^28 - x^27 - 115*x^26 + 115*x^25 + 5917*x^24 - 5917*x^23 - 179683*x^22 + 179683*x^21 + 3576861*x^20 - 3576861*x^19 - 49014755*x^18 + 49014755*x^17 + 472328221*x^16 - 472328221*x^15 - 3210926051*x^14 + 3210926051*x^13 + 15205345309*x^12 - 15205345309*x^11 - 48637728739*x^10 + 48637728739*x^9 + 99209390109*x^8 - 99209390109*x^7 - 115840964579*x^6 + 115840964579*x^5 + 61259327517*x^4 - 61259327517*x^3 - 6856169443*x^2 + 6856169443*x + 928458781);
 
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^28 - x^27 - 115*x^26 + 115*x^25 + 5917*x^24 - 5917*x^23 - 179683*x^22 + 179683*x^21 + 3576861*x^20 - 3576861*x^19 - 49014755*x^18 + 49014755*x^17 + 472328221*x^16 - 472328221*x^15 - 3210926051*x^14 + 3210926051*x^13 + 15205345309*x^12 - 15205345309*x^11 - 48637728739*x^10 + 48637728739*x^9 + 99209390109*x^8 - 99209390109*x^7 - 115840964579*x^6 + 115840964579*x^5 + 61259327517*x^4 - 61259327517*x^3 - 6856169443*x^2 + 6856169443*x + 928458781);
 
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

$C_{28}$ (as 28T1):

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 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.5487525.2, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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 $28$ R R ${\href{/padicField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/padicField/23.7.0.1}{7} }^{4}$ R $28$ $28$ ${\href{/padicField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/padicField/53.7.0.1}{7} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{14}$

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 $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $28$$2$$14$$14$
\(5\) Copy content Toggle raw display 5.14.7.1$x^{14} + 140 x^{13} + 8435 x^{12} + 284200 x^{11} + 5810525 x^{10} + 72852500 x^{9} + 534104381 x^{8} + 1994350486 x^{7} + 2670547075 x^{6} + 1822151870 x^{5} + 743294125 x^{4} + 386790250 x^{3} + 1508497384 x^{2} + 5074882448 x + 4401772109$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.1$x^{14} + 140 x^{13} + 8435 x^{12} + 284200 x^{11} + 5810525 x^{10} + 72852500 x^{9} + 534104381 x^{8} + 1994350486 x^{7} + 2670547075 x^{6} + 1822151870 x^{5} + 743294125 x^{4} + 386790250 x^{3} + 1508497384 x^{2} + 5074882448 x + 4401772109$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
\(29\) Copy content Toggle raw display Deg $28$$28$$1$$27$