Properties

Label 40.0.100...625.2
Degree $40$
Signature $[0, 20]$
Discriminant $1.004\times 10^{68}$
Root discriminant \(50.12\)
Ramified primes $3,5,11$
Class number not computed
Class group not computed
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 5*x^38 - 5*x^37 + 20*x^36 - 19*x^35 + 74*x^34 - 65*x^33 + 265*x^32 - 210*x^31 + 936*x^30 - 1475*x^29 + 4114*x^28 - 6060*x^27 + 15680*x^26 - 21989*x^25 + 56355*x^24 - 75531*x^23 + 197315*x^22 - 252030*x^21 + 682811*x^20 - 826140*x^19 + 1114445*x^18 - 1435731*x^17 + 1906140*x^16 - 2365144*x^15 + 3132335*x^14 - 3532620*x^13 + 4695244*x^12 - 3931240*x^11 + 5449401*x^10 + 1139235*x^9 + 238165*x^8 + 49790*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1)
 
gp: K = bnfinit(y^40 - y^39 + 5*y^38 - 5*y^37 + 20*y^36 - 19*y^35 + 74*y^34 - 65*y^33 + 265*y^32 - 210*y^31 + 936*y^30 - 1475*y^29 + 4114*y^28 - 6060*y^27 + 15680*y^26 - 21989*y^25 + 56355*y^24 - 75531*y^23 + 197315*y^22 - 252030*y^21 + 682811*y^20 - 826140*y^19 + 1114445*y^18 - 1435731*y^17 + 1906140*y^16 - 2365144*y^15 + 3132335*y^14 - 3532620*y^13 + 4695244*y^12 - 3931240*y^11 + 5449401*y^10 + 1139235*y^9 + 238165*y^8 + 49790*y^7 + 10409*y^6 + 2176*y^5 + 455*y^4 + 95*y^3 + 20*y^2 + 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - x^39 + 5*x^38 - 5*x^37 + 20*x^36 - 19*x^35 + 74*x^34 - 65*x^33 + 265*x^32 - 210*x^31 + 936*x^30 - 1475*x^29 + 4114*x^28 - 6060*x^27 + 15680*x^26 - 21989*x^25 + 56355*x^24 - 75531*x^23 + 197315*x^22 - 252030*x^21 + 682811*x^20 - 826140*x^19 + 1114445*x^18 - 1435731*x^17 + 1906140*x^16 - 2365144*x^15 + 3132335*x^14 - 3532620*x^13 + 4695244*x^12 - 3931240*x^11 + 5449401*x^10 + 1139235*x^9 + 238165*x^8 + 49790*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - x^39 + 5*x^38 - 5*x^37 + 20*x^36 - 19*x^35 + 74*x^34 - 65*x^33 + 265*x^32 - 210*x^31 + 936*x^30 - 1475*x^29 + 4114*x^28 - 6060*x^27 + 15680*x^26 - 21989*x^25 + 56355*x^24 - 75531*x^23 + 197315*x^22 - 252030*x^21 + 682811*x^20 - 826140*x^19 + 1114445*x^18 - 1435731*x^17 + 1906140*x^16 - 2365144*x^15 + 3132335*x^14 - 3532620*x^13 + 4695244*x^12 - 3931240*x^11 + 5449401*x^10 + 1139235*x^9 + 238165*x^8 + 49790*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1)
 

\( x^{40} - x^{39} + 5 x^{38} - 5 x^{37} + 20 x^{36} - 19 x^{35} + 74 x^{34} - 65 x^{33} + 265 x^{32} - 210 x^{31} + 936 x^{30} - 1475 x^{29} + 4114 x^{28} - 6060 x^{27} + 15680 x^{26} - 21989 x^{25} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(100383397447978918530459891214693626269465146363712847232818603515625\) \(\medspace = 3^{20}\cdot 5^{30}\cdot 11^{36}\) 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:  \(50.12\)
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:  $3^{1/2}5^{3/4}11^{9/10}\approx 50.12351825429183$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(128,·)$, $\chi_{165}(1,·)$, $\chi_{165}(2,·)$, $\chi_{165}(4,·)$, $\chi_{165}(8,·)$, $\chi_{165}(137,·)$, $\chi_{165}(139,·)$, $\chi_{165}(16,·)$, $\chi_{165}(17,·)$, $\chi_{165}(19,·)$, $\chi_{165}(23,·)$, $\chi_{165}(152,·)$, $\chi_{165}(158,·)$, $\chi_{165}(31,·)$, $\chi_{165}(32,·)$, $\chi_{165}(34,·)$, $\chi_{165}(38,·)$, $\chi_{165}(46,·)$, $\chi_{165}(47,·)$, $\chi_{165}(49,·)$, $\chi_{165}(53,·)$, $\chi_{165}(151,·)$, $\chi_{165}(61,·)$, $\chi_{165}(62,·)$, $\chi_{165}(64,·)$, $\chi_{165}(68,·)$, $\chi_{165}(76,·)$, $\chi_{165}(79,·)$, $\chi_{165}(83,·)$, $\chi_{165}(91,·)$, $\chi_{165}(92,·)$, $\chi_{165}(122,·)$, $\chi_{165}(94,·)$, $\chi_{165}(98,·)$, $\chi_{165}(106,·)$, $\chi_{165}(107,·)$, $\chi_{165}(109,·)$, $\chi_{165}(113,·)$, $\chi_{165}(136,·)$, $\chi_{165}(124,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{25\!\cdots\!21}a^{31}+\frac{969862058041036}{25\!\cdots\!21}a^{30}+\frac{241303777751570}{25\!\cdots\!21}a^{29}+\frac{11\!\cdots\!57}{25\!\cdots\!21}a^{28}+\frac{12\!\cdots\!45}{25\!\cdots\!21}a^{27}+\frac{710382005693106}{25\!\cdots\!21}a^{26}+\frac{743525622606569}{25\!\cdots\!21}a^{25}+\frac{759727505551736}{25\!\cdots\!21}a^{24}-\frac{12\!\cdots\!23}{25\!\cdots\!21}a^{23}-\frac{10\!\cdots\!26}{25\!\cdots\!21}a^{22}+\frac{10\!\cdots\!30}{25\!\cdots\!21}a^{21}+\frac{95503815941798}{25\!\cdots\!21}a^{20}+\frac{25370031947195}{25\!\cdots\!21}a^{19}-\frac{528579085352603}{25\!\cdots\!21}a^{18}-\frac{70364038799755}{25\!\cdots\!21}a^{17}+\frac{488230004469246}{25\!\cdots\!21}a^{16}-\frac{383166537792952}{25\!\cdots\!21}a^{15}+\frac{57003490590598}{25\!\cdots\!21}a^{14}+\frac{433614414914333}{25\!\cdots\!21}a^{13}+\frac{299709387039926}{25\!\cdots\!21}a^{12}-\frac{480277222460171}{25\!\cdots\!21}a^{11}+\frac{830362944453688}{25\!\cdots\!21}a^{10}-\frac{14059251761729}{25\!\cdots\!21}a^{9}-\frac{484226564561075}{25\!\cdots\!21}a^{8}+\frac{126960133769841}{25\!\cdots\!21}a^{7}-\frac{424260348281824}{25\!\cdots\!21}a^{6}-\frac{990746123121383}{25\!\cdots\!21}a^{5}+\frac{285771829634526}{25\!\cdots\!21}a^{4}-\frac{10\!\cdots\!53}{25\!\cdots\!21}a^{3}+\frac{11\!\cdots\!70}{25\!\cdots\!21}a^{2}-\frac{572259025431674}{25\!\cdots\!21}a+\frac{808763874066416}{25\!\cdots\!21}$, $\frac{1}{25\!\cdots\!21}a^{32}+\frac{12\!\cdots\!25}{25\!\cdots\!21}a^{30}-\frac{613072852992611}{25\!\cdots\!21}a^{29}+\frac{503379744866868}{25\!\cdots\!21}a^{28}-\frac{539158269730133}{25\!\cdots\!21}a^{27}-\frac{11\!\cdots\!64}{25\!\cdots\!21}a^{26}-\frac{920953358335506}{25\!\cdots\!21}a^{25}-\frac{12\!\cdots\!08}{25\!\cdots\!21}a^{24}+\frac{11\!\cdots\!24}{25\!\cdots\!21}a^{23}-\frac{10\!\cdots\!53}{25\!\cdots\!21}a^{22}-\frac{95503815942622}{25\!\cdots\!21}a^{21}+\frac{209142208610269}{25\!\cdots\!21}a^{20}+\frac{639739986408395}{25\!\cdots\!21}a^{19}-\frac{693236095240533}{25\!\cdots\!21}a^{18}-\frac{868143299208031}{25\!\cdots\!21}a^{17}+\frac{445016655269210}{25\!\cdots\!21}a^{16}+\frac{248343766226902}{25\!\cdots\!21}a^{15}-\frac{12\!\cdots\!53}{25\!\cdots\!21}a^{14}-\frac{163212758531710}{25\!\cdots\!21}a^{13}+\frac{773526171669338}{25\!\cdots\!21}a^{12}+\frac{913440026709071}{25\!\cdots\!21}a^{11}-\frac{304794668595476}{25\!\cdots\!21}a^{10}-\frac{916762311003590}{25\!\cdots\!21}a^{9}-\frac{10\!\cdots\!44}{25\!\cdots\!21}a^{8}+\frac{23573903759522}{25\!\cdots\!21}a^{7}-\frac{201973183291819}{25\!\cdots\!21}a^{6}-\frac{655525585245758}{25\!\cdots\!21}a^{5}+\frac{10\!\cdots\!14}{25\!\cdots\!21}a^{4}+\frac{558223872986398}{25\!\cdots\!21}a^{3}+\frac{11\!\cdots\!45}{25\!\cdots\!21}a^{2}+\frac{814362117403119}{25\!\cdots\!21}a+\frac{61996354910397}{25\!\cdots\!21}$, $\frac{1}{25\!\cdots\!21}a^{33}+\frac{936795771304776}{25\!\cdots\!21}a^{22}-\frac{554016545945649}{25\!\cdots\!21}a^{11}+\frac{204895633579702}{25\!\cdots\!21}$, $\frac{1}{25\!\cdots\!21}a^{34}+\frac{936795771304776}{25\!\cdots\!21}a^{23}-\frac{554016545945649}{25\!\cdots\!21}a^{12}+\frac{204895633579702}{25\!\cdots\!21}a$, $\frac{1}{25\!\cdots\!21}a^{35}+\frac{936795771304776}{25\!\cdots\!21}a^{24}-\frac{554016545945649}{25\!\cdots\!21}a^{13}+\frac{204895633579702}{25\!\cdots\!21}a^{2}$, $\frac{1}{25\!\cdots\!21}a^{36}+\frac{936795771304776}{25\!\cdots\!21}a^{25}-\frac{554016545945649}{25\!\cdots\!21}a^{14}+\frac{204895633579702}{25\!\cdots\!21}a^{3}$, $\frac{1}{25\!\cdots\!21}a^{37}+\frac{936795771304776}{25\!\cdots\!21}a^{26}-\frac{554016545945649}{25\!\cdots\!21}a^{15}+\frac{204895633579702}{25\!\cdots\!21}a^{4}$, $\frac{1}{25\!\cdots\!21}a^{38}+\frac{936795771304776}{25\!\cdots\!21}a^{27}-\frac{554016545945649}{25\!\cdots\!21}a^{16}+\frac{204895633579702}{25\!\cdots\!21}a^{5}$, $\frac{1}{25\!\cdots\!21}a^{39}+\frac{936795771304776}{25\!\cdots\!21}a^{28}-\frac{554016545945649}{25\!\cdots\!21}a^{17}+\frac{204895633579702}{25\!\cdots\!21}a^{6}$ 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:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{1008753747971}{2526205995232921} a^{39} - \frac{831212771199384}{2526205995232921} a^{28} - \frac{1248337057599278259}{2526205995232921} a^{17} - \frac{30260868834152543254}{2526205995232921} a^{6} \)  (order $22$) 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^40 - x^39 + 5*x^38 - 5*x^37 + 20*x^36 - 19*x^35 + 74*x^34 - 65*x^33 + 265*x^32 - 210*x^31 + 936*x^30 - 1475*x^29 + 4114*x^28 - 6060*x^27 + 15680*x^26 - 21989*x^25 + 56355*x^24 - 75531*x^23 + 197315*x^22 - 252030*x^21 + 682811*x^20 - 826140*x^19 + 1114445*x^18 - 1435731*x^17 + 1906140*x^16 - 2365144*x^15 + 3132335*x^14 - 3532620*x^13 + 4695244*x^12 - 3931240*x^11 + 5449401*x^10 + 1139235*x^9 + 238165*x^8 + 49790*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*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^40 - x^39 + 5*x^38 - 5*x^37 + 20*x^36 - 19*x^35 + 74*x^34 - 65*x^33 + 265*x^32 - 210*x^31 + 936*x^30 - 1475*x^29 + 4114*x^28 - 6060*x^27 + 15680*x^26 - 21989*x^25 + 56355*x^24 - 75531*x^23 + 197315*x^22 - 252030*x^21 + 682811*x^20 - 826140*x^19 + 1114445*x^18 - 1435731*x^17 + 1906140*x^16 - 2365144*x^15 + 3132335*x^14 - 3532620*x^13 + 4695244*x^12 - 3931240*x^11 + 5449401*x^10 + 1139235*x^9 + 238165*x^8 + 49790*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*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^40 - x^39 + 5*x^38 - 5*x^37 + 20*x^36 - 19*x^35 + 74*x^34 - 65*x^33 + 265*x^32 - 210*x^31 + 936*x^30 - 1475*x^29 + 4114*x^28 - 6060*x^27 + 15680*x^26 - 21989*x^25 + 56355*x^24 - 75531*x^23 + 197315*x^22 - 252030*x^21 + 682811*x^20 - 826140*x^19 + 1114445*x^18 - 1435731*x^17 + 1906140*x^16 - 2365144*x^15 + 3132335*x^14 - 3532620*x^13 + 4695244*x^12 - 3931240*x^11 + 5449401*x^10 + 1139235*x^9 + 238165*x^8 + 49790*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*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^40 - x^39 + 5*x^38 - 5*x^37 + 20*x^36 - 19*x^35 + 74*x^34 - 65*x^33 + 265*x^32 - 210*x^31 + 936*x^30 - 1475*x^29 + 4114*x^28 - 6060*x^27 + 15680*x^26 - 21989*x^25 + 56355*x^24 - 75531*x^23 + 197315*x^22 - 252030*x^21 + 682811*x^20 - 826140*x^19 + 1114445*x^18 - 1435731*x^17 + 1906140*x^16 - 2365144*x^15 + 3132335*x^14 - 3532620*x^13 + 4695244*x^12 - 3931240*x^11 + 5449401*x^10 + 1139235*x^9 + 238165*x^8 + 49790*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*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

$C_2\times C_{20}$ (as 40T2):

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)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.0.136125.2, \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{11})^+\), 8.0.18530015625.1, \(\Q(\zeta_{11})\), 10.10.669871503125.1, 10.0.7368586534375.1, 20.0.54296067514572573056640625.1, 20.0.10019151533337487082567413330078125.1, 20.20.82802905234194108120391845703125.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $20^{2}$ R R $20^{2}$ R $20^{2}$ $20^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{8}$

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 3.20.10.2$x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$$2$$10$$10$20T1$[\ ]_{2}^{10}$
3.20.10.2$x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$$2$$10$$10$20T1$[\ ]_{2}^{10}$
\(5\) Copy content Toggle raw display Deg $20$$4$$5$$15$
Deg $20$$4$$5$$15$
\(11\) Copy content Toggle raw display 11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$