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Properties

Label 44.0.782...952.1
Degree $44$
Signature $[0, 22]$
Discriminant $7.830\times 10^{90}$
Root discriminant \(116.35\)
Ramified primes $2,23$
Class number not computed
Class group not computed
Galois group $C_{44}$ (as 44T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 84*x^42 + 3162*x^40 + 70680*x^38 + 1048936*x^36 + 10957792*x^34 + 83449416*x^32 + 473860064*x^30 + 2036712064*x^28 + 6691379968*x^26 + 16898234240*x^24 + 32860916224*x^22 + 49105446400*x^20 + 56035526656*x^18 + 48285946240*x^16 + 30880705024*x^14 + 14289749760*x^12 + 4610997248*x^10 + 982631936*x^8 + 127395840*x^6 + 8853504*x^4 + 270336*x^2 + 2048)
 
gp: K = bnfinit(y^44 + 84*y^42 + 3162*y^40 + 70680*y^38 + 1048936*y^36 + 10957792*y^34 + 83449416*y^32 + 473860064*y^30 + 2036712064*y^28 + 6691379968*y^26 + 16898234240*y^24 + 32860916224*y^22 + 49105446400*y^20 + 56035526656*y^18 + 48285946240*y^16 + 30880705024*y^14 + 14289749760*y^12 + 4610997248*y^10 + 982631936*y^8 + 127395840*y^6 + 8853504*y^4 + 270336*y^2 + 2048, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 + 84*x^42 + 3162*x^40 + 70680*x^38 + 1048936*x^36 + 10957792*x^34 + 83449416*x^32 + 473860064*x^30 + 2036712064*x^28 + 6691379968*x^26 + 16898234240*x^24 + 32860916224*x^22 + 49105446400*x^20 + 56035526656*x^18 + 48285946240*x^16 + 30880705024*x^14 + 14289749760*x^12 + 4610997248*x^10 + 982631936*x^8 + 127395840*x^6 + 8853504*x^4 + 270336*x^2 + 2048);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 + 84*x^42 + 3162*x^40 + 70680*x^38 + 1048936*x^36 + 10957792*x^34 + 83449416*x^32 + 473860064*x^30 + 2036712064*x^28 + 6691379968*x^26 + 16898234240*x^24 + 32860916224*x^22 + 49105446400*x^20 + 56035526656*x^18 + 48285946240*x^16 + 30880705024*x^14 + 14289749760*x^12 + 4610997248*x^10 + 982631936*x^8 + 127395840*x^6 + 8853504*x^4 + 270336*x^2 + 2048)
 

\( x^{44} + 84 x^{42} + 3162 x^{40} + 70680 x^{38} + 1048936 x^{36} + 10957792 x^{34} + 83449416 x^{32} + \cdots + 2048 \) Copy content Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(782\!\cdots\!952\) \(\medspace = 2^{121}\cdot 23^{40}\) 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:  \(116.35\)
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:  $2^{11/4}23^{10/11}\approx 116.35013310446064$
Ramified primes:   \(2\), \(23\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(368=2^{4}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{368}(1,·)$, $\chi_{368}(259,·)$, $\chi_{368}(257,·)$, $\chi_{368}(9,·)$, $\chi_{368}(139,·)$, $\chi_{368}(3,·)$, $\chi_{368}(25,·)$, $\chi_{368}(27,·)$, $\chi_{368}(289,·)$, $\chi_{368}(35,·)$, $\chi_{368}(163,·)$, $\chi_{368}(49,·)$, $\chi_{368}(41,·)$, $\chi_{368}(307,·)$, $\chi_{368}(177,·)$, $\chi_{368}(179,·)$, $\chi_{368}(265,·)$, $\chi_{368}(185,·)$, $\chi_{368}(347,·)$, $\chi_{368}(59,·)$, $\chi_{368}(193,·)$, $\chi_{368}(323,·)$, $\chi_{368}(147,·)$, $\chi_{368}(353,·)$, $\chi_{368}(73,·)$, $\chi_{368}(75,·)$, $\chi_{368}(305,·)$, $\chi_{368}(81,·)$, $\chi_{368}(211,·)$, $\chi_{368}(105,·)$, $\chi_{368}(169,·)$, $\chi_{368}(331,·)$, $\chi_{368}(219,·)$, $\chi_{368}(225,·)$, $\chi_{368}(315,·)$, $\chi_{368}(131,·)$, $\chi_{368}(209,·)$, $\chi_{368}(233,·)$, $\chi_{368}(363,·)$, $\chi_{368}(243,·)$, $\chi_{368}(187,·)$, $\chi_{368}(361,·)$, $\chi_{368}(121,·)$, $\chi_{368}(123,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{16}a^{18}$, $\frac{1}{16}a^{19}$, $\frac{1}{32}a^{20}$, $\frac{1}{32}a^{21}$, $\frac{1}{32}a^{22}$, $\frac{1}{32}a^{23}$, $\frac{1}{64}a^{24}$, $\frac{1}{64}a^{25}$, $\frac{1}{64}a^{26}$, $\frac{1}{64}a^{27}$, $\frac{1}{128}a^{28}$, $\frac{1}{128}a^{29}$, $\frac{1}{128}a^{30}$, $\frac{1}{128}a^{31}$, $\frac{1}{256}a^{32}$, $\frac{1}{256}a^{33}$, $\frac{1}{12032}a^{34}-\frac{9}{6016}a^{32}-\frac{5}{3008}a^{30}-\frac{11}{3008}a^{28}+\frac{1}{752}a^{26}-\frac{5}{752}a^{24}-\frac{15}{1504}a^{22}+\frac{23}{1504}a^{20}+\frac{21}{752}a^{18}+\frac{3}{188}a^{16}-\frac{13}{376}a^{14}+\frac{3}{94}a^{12}+\frac{7}{188}a^{10}-\frac{3}{188}a^{8}-\frac{15}{94}a^{6}-\frac{11}{47}a^{4}-\frac{15}{47}a^{2}-\frac{19}{47}$, $\frac{1}{12032}a^{35}-\frac{9}{6016}a^{33}-\frac{5}{3008}a^{31}-\frac{11}{3008}a^{29}+\frac{1}{752}a^{27}-\frac{5}{752}a^{25}-\frac{15}{1504}a^{23}+\frac{23}{1504}a^{21}+\frac{21}{752}a^{19}+\frac{3}{188}a^{17}-\frac{13}{376}a^{15}+\frac{3}{94}a^{13}+\frac{7}{188}a^{11}-\frac{3}{188}a^{9}-\frac{15}{94}a^{7}-\frac{11}{47}a^{5}-\frac{15}{47}a^{3}-\frac{19}{47}a$, $\frac{1}{24064}a^{36}+\frac{1}{752}a^{32}-\frac{7}{6016}a^{30}-\frac{3}{3008}a^{28}-\frac{21}{3008}a^{26}-\frac{7}{3008}a^{24}-\frac{3}{752}a^{22}-\frac{7}{1504}a^{20}+\frac{7}{752}a^{18}+\frac{1}{752}a^{16}-\frac{17}{376}a^{14}+\frac{21}{376}a^{12}-\frac{9}{188}a^{10}+\frac{5}{188}a^{8}-\frac{5}{94}a^{6}+\frac{11}{47}a^{4}+\frac{20}{47}a^{2}+\frac{17}{47}$, $\frac{1}{24064}a^{37}+\frac{1}{752}a^{33}-\frac{7}{6016}a^{31}-\frac{3}{3008}a^{29}-\frac{21}{3008}a^{27}-\frac{7}{3008}a^{25}-\frac{3}{752}a^{23}-\frac{7}{1504}a^{21}+\frac{7}{752}a^{19}+\frac{1}{752}a^{17}-\frac{17}{376}a^{15}+\frac{21}{376}a^{13}-\frac{9}{188}a^{11}+\frac{5}{188}a^{9}-\frac{5}{94}a^{7}+\frac{11}{47}a^{5}+\frac{20}{47}a^{3}+\frac{17}{47}a$, $\frac{1}{24064}a^{38}-\frac{1}{1504}a^{32}+\frac{13}{6016}a^{30}-\frac{19}{6016}a^{28}+\frac{23}{3008}a^{26}-\frac{21}{3008}a^{24}-\frac{1}{752}a^{22}+\frac{11}{752}a^{20}-\frac{3}{376}a^{18}+\frac{9}{752}a^{16}-\frac{3}{188}a^{14}-\frac{11}{188}a^{12}-\frac{13}{188}a^{10}-\frac{9}{188}a^{8}-\frac{10}{47}a^{6}+\frac{8}{47}a^{4}+\frac{22}{47}a^{2}+\frac{22}{47}$, $\frac{1}{24064}a^{39}-\frac{1}{1504}a^{33}+\frac{13}{6016}a^{31}-\frac{19}{6016}a^{29}+\frac{23}{3008}a^{27}-\frac{21}{3008}a^{25}-\frac{1}{752}a^{23}+\frac{11}{752}a^{21}-\frac{3}{376}a^{19}+\frac{9}{752}a^{17}-\frac{3}{188}a^{15}-\frac{11}{188}a^{13}-\frac{13}{188}a^{11}-\frac{9}{188}a^{9}-\frac{10}{47}a^{7}+\frac{8}{47}a^{5}+\frac{22}{47}a^{3}+\frac{22}{47}a$, $\frac{1}{6593536}a^{40}-\frac{61}{3296768}a^{38}+\frac{21}{3296768}a^{36}-\frac{9}{824192}a^{34}-\frac{419}{824192}a^{32}+\frac{547}{412096}a^{30}-\frac{2913}{824192}a^{28}+\frac{181}{412096}a^{26}-\frac{91}{12878}a^{24}-\frac{1381}{206048}a^{22}+\frac{24}{6439}a^{20}-\frac{2291}{103024}a^{18}-\frac{387}{25756}a^{16}-\frac{1747}{51512}a^{14}-\frac{701}{51512}a^{12}+\frac{333}{12878}a^{10}-\frac{489}{6439}a^{8}+\frac{111}{12878}a^{6}+\frac{1649}{12878}a^{4}-\frac{2440}{6439}a^{2}-\frac{484}{6439}$, $\frac{1}{6593536}a^{41}-\frac{61}{3296768}a^{39}+\frac{21}{3296768}a^{37}-\frac{9}{824192}a^{35}-\frac{419}{824192}a^{33}+\frac{547}{412096}a^{31}-\frac{2913}{824192}a^{29}+\frac{181}{412096}a^{27}-\frac{91}{12878}a^{25}-\frac{1381}{206048}a^{23}+\frac{24}{6439}a^{21}-\frac{2291}{103024}a^{19}-\frac{387}{25756}a^{17}-\frac{1747}{51512}a^{15}-\frac{701}{51512}a^{13}+\frac{333}{12878}a^{11}-\frac{489}{6439}a^{9}+\frac{111}{12878}a^{7}+\frac{1649}{12878}a^{5}-\frac{2440}{6439}a^{3}-\frac{484}{6439}a$, $\frac{1}{18\!\cdots\!44}a^{42}-\frac{90\!\cdots\!37}{18\!\cdots\!44}a^{40}-\frac{68\!\cdots\!05}{47\!\cdots\!36}a^{38}+\frac{14\!\cdots\!85}{94\!\cdots\!72}a^{36}-\frac{15\!\cdots\!53}{11\!\cdots\!84}a^{34}+\frac{19\!\cdots\!21}{47\!\cdots\!36}a^{32}+\frac{64\!\cdots\!13}{23\!\cdots\!68}a^{30}+\frac{51\!\cdots\!79}{23\!\cdots\!68}a^{28}-\frac{60\!\cdots\!97}{11\!\cdots\!84}a^{26}-\frac{30\!\cdots\!99}{58\!\cdots\!92}a^{24}+\frac{77\!\cdots\!17}{58\!\cdots\!92}a^{22}-\frac{26\!\cdots\!53}{58\!\cdots\!92}a^{20}+\frac{17\!\cdots\!71}{14\!\cdots\!48}a^{18}+\frac{11\!\cdots\!33}{36\!\cdots\!62}a^{16}-\frac{48\!\cdots\!25}{14\!\cdots\!48}a^{14}-\frac{21\!\cdots\!15}{36\!\cdots\!62}a^{12}+\frac{21\!\cdots\!23}{73\!\cdots\!24}a^{10}-\frac{31\!\cdots\!27}{73\!\cdots\!24}a^{8}+\frac{24\!\cdots\!71}{18\!\cdots\!81}a^{6}-\frac{41\!\cdots\!99}{36\!\cdots\!62}a^{4}-\frac{16\!\cdots\!87}{18\!\cdots\!81}a^{2}-\frac{91\!\cdots\!27}{18\!\cdots\!81}$, $\frac{1}{18\!\cdots\!44}a^{43}-\frac{90\!\cdots\!37}{18\!\cdots\!44}a^{41}-\frac{68\!\cdots\!05}{47\!\cdots\!36}a^{39}+\frac{14\!\cdots\!85}{94\!\cdots\!72}a^{37}-\frac{15\!\cdots\!53}{11\!\cdots\!84}a^{35}+\frac{19\!\cdots\!21}{47\!\cdots\!36}a^{33}+\frac{64\!\cdots\!13}{23\!\cdots\!68}a^{31}+\frac{51\!\cdots\!79}{23\!\cdots\!68}a^{29}-\frac{60\!\cdots\!97}{11\!\cdots\!84}a^{27}-\frac{30\!\cdots\!99}{58\!\cdots\!92}a^{25}+\frac{77\!\cdots\!17}{58\!\cdots\!92}a^{23}-\frac{26\!\cdots\!53}{58\!\cdots\!92}a^{21}+\frac{17\!\cdots\!71}{14\!\cdots\!48}a^{19}+\frac{11\!\cdots\!33}{36\!\cdots\!62}a^{17}-\frac{48\!\cdots\!25}{14\!\cdots\!48}a^{15}-\frac{21\!\cdots\!15}{36\!\cdots\!62}a^{13}+\frac{21\!\cdots\!23}{73\!\cdots\!24}a^{11}-\frac{31\!\cdots\!27}{73\!\cdots\!24}a^{9}+\frac{24\!\cdots\!71}{18\!\cdots\!81}a^{7}-\frac{41\!\cdots\!99}{36\!\cdots\!62}a^{5}-\frac{16\!\cdots\!87}{18\!\cdots\!81}a^{3}-\frac{91\!\cdots\!27}{18\!\cdots\!81}a$ 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:  $21$
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^44 + 84*x^42 + 3162*x^40 + 70680*x^38 + 1048936*x^36 + 10957792*x^34 + 83449416*x^32 + 473860064*x^30 + 2036712064*x^28 + 6691379968*x^26 + 16898234240*x^24 + 32860916224*x^22 + 49105446400*x^20 + 56035526656*x^18 + 48285946240*x^16 + 30880705024*x^14 + 14289749760*x^12 + 4610997248*x^10 + 982631936*x^8 + 127395840*x^6 + 8853504*x^4 + 270336*x^2 + 2048)
 
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^44 + 84*x^42 + 3162*x^40 + 70680*x^38 + 1048936*x^36 + 10957792*x^34 + 83449416*x^32 + 473860064*x^30 + 2036712064*x^28 + 6691379968*x^26 + 16898234240*x^24 + 32860916224*x^22 + 49105446400*x^20 + 56035526656*x^18 + 48285946240*x^16 + 30880705024*x^14 + 14289749760*x^12 + 4610997248*x^10 + 982631936*x^8 + 127395840*x^6 + 8853504*x^4 + 270336*x^2 + 2048, 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^44 + 84*x^42 + 3162*x^40 + 70680*x^38 + 1048936*x^36 + 10957792*x^34 + 83449416*x^32 + 473860064*x^30 + 2036712064*x^28 + 6691379968*x^26 + 16898234240*x^24 + 32860916224*x^22 + 49105446400*x^20 + 56035526656*x^18 + 48285946240*x^16 + 30880705024*x^14 + 14289749760*x^12 + 4610997248*x^10 + 982631936*x^8 + 127395840*x^6 + 8853504*x^4 + 270336*x^2 + 2048);
 
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^44 + 84*x^42 + 3162*x^40 + 70680*x^38 + 1048936*x^36 + 10957792*x^34 + 83449416*x^32 + 473860064*x^30 + 2036712064*x^28 + 6691379968*x^26 + 16898234240*x^24 + 32860916224*x^22 + 49105446400*x^20 + 56035526656*x^18 + 48285946240*x^16 + 30880705024*x^14 + 14289749760*x^12 + 4610997248*x^10 + 982631936*x^8 + 127395840*x^6 + 8853504*x^4 + 270336*x^2 + 2048);
 
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

$C_{44}$ (as 44T1):

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 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.2048.2, \(\Q(\zeta_{23})^+\), 22.22.14741666340843480753092741810452692992.1

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 R $44$ $44$ ${\href{/padicField/7.11.0.1}{11} }^{4}$ $44$ $44$ ${\href{/padicField/17.11.0.1}{11} }^{4}$ $44$ R $44$ $22^{2}$ $44$ $22^{2}$ $44$ ${\href{/padicField/47.2.0.1}{2} }^{22}$ $44$ $44$

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
\(2\) Copy content Toggle raw display Deg $44$$4$$11$$121$
\(23\) Copy content Toggle raw display 23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} + 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$