# $p$-adic families downloaded from the LMFDB on 17 July 2026. # Search link: https://www.lmfdb.org/padicField/?relative=1&search_type=Families&label_absolute=17.3.2.3a&base=17.1.1.0a1.1,17.1.2.1a1.1,17.3.1.0a1.1 # Query "{'base': {'$in': ['17.1.1.0a1.1', '17.1.2.1a1.1', '17.3.1.0a1.1']}, 'label_absolute': '17.3.2.3a'}" returned 3 families, sorted by base. # Each entry in the following data list has the form: # [Label, $p$, $n$, $f$, $e$, $c$, Base, Swan slopes, Means, Rams, Ambiguity, Field count, Mass] # For more details, see the definitions at the bottom of the file. "17.3.2.3a" 17 6 3 2 3 "17.1.1.0a1.1" [] [] [] 6 2 "1" "17.1.2.1a1.1-3.1.0a" 17 3 3 1 0 "17.1.2.1a1.1" [] [] [] 3 1 "1" "17.3.1.0a1.1-1.2.1a" 17 2 1 2 1 "17.3.1.0a1.1" [] [] [] 2 2 "1" # Label -- # Let $K$ be a finite extension of $\Q_p$, with $K\ne\Q_p$. # The label associated to a relative # family $I/K$ has the form # $s\text{-}f.e.c\ell$, where # - $s$ is the label attached to the base # field $K$, # - $f$ is the residue field degree # of every $L/K\in I/K$, # - $e$ is the ramification index of # every $L/K\in I/K$, # - $c$ is the discriminant exponent # of every $L/K\in I/K$, # - $\ell$ is a string of one or more letters used to distinguish families # with the same basic data. It is defined by sorting the families by their number of wild segments, then the list of lengths of the wild segments, then the rams. # The label associated to an absolute family $I/\Q_p$ has the form # $p.f.e.c\ell$, where $f,e,c,\ell$ are defined as above, with $K=\Q_p$. #$p$ (p) -- # The **residue field** of a nonarchimedean local field is the quotient of its ring of integers by its unique maximal ideal. # The residue field is finite and its characteristic $p$ is the **residue field characteristic**. Finite extensions of $\Q_p$ have residue field characteristic $p$. #$n$ (n) -- # Let $L/K$ be an extension of $p$-adic fields. # The **degree** of $L/K$ is the dimension of $L$ as a vector space over $K$. # The **base degree** $n_0$ of $L/K$ is the degree of $K/\Q_p$. The **absolute degree** $n_{\mathrm{abs}}$ # of $L/K$ is the degree of $L/\Q_p$. #$f$ (f) -- # Let $L/K$ be an extension of $p$-adic fields. # Let $\kappa$ be the residue field of $K$ # and let $\lambda$ be the residue field of $L$. The **residue field degree** $f$ # of $L/K$ is the degree of the field extension $\lambda/\kappa$. # The **base residue field degree** $f_0$ of $L/K$ is the residue field degree of $K/\Q_p$. The **absolute residue field degree** $f_{\mathrm{abs}}$ of $L/K$ is the # residue field degree of $L$ over $\Q_p$. #$e$ (e) -- # Let $L/K$ be a finite extension of $p$-adic fields. # Then $\mathcal{O}_L$ and $\mathcal{O}_K$, the rings of integers of $L$ and $K$, # are discrete valuation domains, so they have unique maximal ideals $P_L$ and # $P_K$ which are principal. If $P_K=(\pi_K)$, the element $\pi_K$ is a # **uniformizer** for $K$. # We have $\pi_K\mathcal{O}_L=P_L^e$ for some positive integer $e$, which factors as $e = p^w e_{\mathrm{tame}}$ with $e_{\mathrm{tame}}$ coprime to $p$. The integer $e$ # is the **ramification index**, $w$ is the **wild ramification exponent** and $e_{\mathrm{tame}}$ is the **tame ramification index** of $L/K$. # - If $e=1$ we say that the extension $L/K$ is **unramified**. # - If $e=[L:K]$ we say that $L/K$ is **totally ramified**. # - If $p \nmid e$ we say that $L/K$ is **tamely ramified**. # The **base ramification index** $e_0$ of $L/K$ is the ramification index of $K/\Q_p$. # The **absolute ramification index** $e_{\mathrm{abs}}$ of $L/K$ is the ramification index of # $L$ over $\Q_p$. #$c$ (c) -- # Let $L/K$ be a finite extension of $p$-adic fields, with rings of integers $\mathcal{O}_L$ and $\mathcal{O}_K$ and uniformizers $\pi_L$ and $\pi_K$. The **discriminant** of $L/K$ is the square of the determinant of the matrix # \[ # \left( \begin{array}{ccc} # \sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\ # \vdots & & \vdots \\ # \sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n) \\ # \end{array} \right) # \] # where $\sigma_1,..., \sigma_n$ are the embeddings of $L$ into an algebraic closure $\overline{K}$, and $\{\beta_1, \ldots, \beta_n\}$ is a basis for $\mathcal{O}_L$ as a free $\mathcal{O}_K$-module. # The discriminant of $L/K$ is an element of $K^\times$ which is well-defined up to the square of a unit. Thus, it is of the form $\pi_K^c u$ where $u \in \mathcal{O}_K^\times$ is a unit. The value $c$ is the **discriminant exponent** for $L/K$. Together with the discriminant root field of $L/K$, it determines the discriminant of $L/K$ (up to the square of a unit). # The **base discriminant exponent** $c_0$ of $L/K$ is the discriminant exponent of $K/\Q_p$. The **absolute discriminant exponent** $c_{\mathrm{abs}}$ of $L/K$ is the discriminant exponent of $L/\Q_p$. #Base (base_field) -- # A family of extensions of # $p$-adic fields is a collection of # extensions $L/K$ with $K$ a fixed $p$-adic field. The field $K$ # is called the **base field** of the family. #Swan slopes (slopes) -- # Let $L/K$ be an extension of $p$-adic fields. # For each subextension $E/K$ of $L/K$ we plot the point \((n_E,c_E)\), # where $n_E=[E:K]$ and $c_E$ is the # discriminant exponent of $E/K$. # Let $B$ be the boundary of the lower convex hull of these points. The slopes # of the segments of $B$ are called the **Artin slopes** of # $L/K$, and those which are greater than 1 are the **wild Artin slopes**. If a segment corresponding to a wild slope runs from \((n_1, c_1)\) to \((n_2, c_2)\) then $n_2/n_1=p^m$ # for some $m\in\N$, and the corresponding Artin slope is repeated $m$ # times. The **Swan slopes** of $L/K$ are obtained by subtracting $1$ from each Artin slope, and the **wild Swan slopes** are precisely the positive ones. The Artin slopes and Swan # slopes of $L/K$ are also referred to as the **visible Artin slopes** and the # **visible Swan slopes**. This is to distinguish them from the # hidden slopes of $L/K$. # Let $f$ be the residue field degree, let $\epsilon$ be the tame degree of $L/K$, and let $a_1\le a_2\le\dots\le a_w$ be the wild Artin slopes of $L/K$. # The **Artin slope content** of $L/K$ is the collection of data # $[a_1,\dots,a_w]_{\epsilon}^f$. The **Swan slope content** of $L/K$ is $[s_1,\dots,s_w]_{\epsilon}^f$, where $s_1,\dots,s_w$ are the wild Swan slopes of # $L/K$. # The Swan slopes are the same as the positive upper ramification jumps of # $L/K$. In the case where $L/K$ is a Galois extension these are defined for # instance in Chapter IV of Serre's Local Fields # \cite{doi:10.1007/978-1-4757-5673-9,MR:0554237}. For the general case where # $L/K$ is separable but not necessarily Galois see the appendix to Deligne # \cite{MR:0771673}. # Means -- # Let $L/K$ be an extension of $p$-adic fields. # The **mean** $m$ of $L/K$ is determined by the formula $c=f(e−1+em)$, where # $c$ is the discriminant exponent, # $f$ is the residue field degree, and # $e$ is the ramification index of $L/K$. # The mean of $L/K$ is related to the height # $H(L/K)$ by the formula $H(L/K)=em$. # Let $K_f/K$ be the maximal unramified subextension of $L/K$. For each # subextension $L'/K_f$ of $L/K_f$ set $e'=[L':K_f]$ and # let $m'$ be the mean of $L'/K_f$. We associate to $L'$ the point # $(e',e'm')$ in the Cartesian plane. The **slope polygon** $S$ of $L/K$ is the # lower boundary of the convex hull of these points. Write # $e=\epsilon p^w=[L:K_f]$ and let $h$ be the function on the interval # $[1,e]$ whose graph is $S$. For $1\le k\le w$ set # $m_k=h(\epsilon p^k)/(\epsilon p^k)$. Then the **means** of $L/K$ are defined # to be $\langle m_1,\dots,m_w\rangle$. # If we adopt the convention $m_0=0$ then the means are related to the # Swan slopes $[s_1,\dots,s_w]$ by the formulas # $(p-1)s_k=pm_k-m_{k-1}$ for $1\le k\le w$. # Rams -- # Let $L/K$ be an extension of $p$-adic fields # with ramification index $e = \epsilon p^w$. # The **rams** $r_1 \le \dots \le r_w$ of $L/K$ are the same as the positive lower # ramification jumps of $L/K$. In the case where $L/K$ is a Galois extension the # lower jumps are defined in Chapter IV of Serre's Local Fields # \cite{doi:10.1007/978-1-4757-5673-9,MR:0554237}. # For the general case where $L/K$ is separable but not necessarily # Galois see the appendix to Deligne \cite{MR:0771673}. When $L/K$ is Galois a # ram is any number of the form $v_L(\sigma(\pi_L)-\pi_L)-1$ for some # $\sigma\not=1_L$ in the wild inertia group of # $\Gal(L/K)$. Thus the rams of a Galois extension are positive integers, but in general a ram may have denominator dividing $p^k-1$ where $k$ is the multiplicity of the ram. # Two rescalings of the rams can be helpful in various contexts. The **small rams** $r_k^*$ and **tiny rams** $r_k'$ are defined by # $$r_k^* = \frac{(p-1)r_k}{\epsilon p^k} \qquad\qquad\qquad r_k' = \frac{r_k}{\epsilon p^k}.$$ # With the convention that $r_0=s_0=m_0=0$, the relationship with the means $\langle m_1, \dots, m_w\rangle$ and Swan slopes $[s_1, \dots, s_w]$ is then given by the equations # $$r_k^* = m_k-m_{k-1} \qquad\qquad\qquad r'_k = s_k-m_k.$$ # Rams are helpful in enumerating the possible Herbrand invariants over $K$. Define first $\mathcal{R}_k^\infty$ to be the set of positive rational numbers with denominator dividing $p^k-1$ and numerator prime to $p$. Then set # $$\mathcal{R}_k^{z} = \left\{\alpha \in \mathcal{R}_k^\infty : \alpha < \frac{pz}{p-1}\right\} \ \ \text{if}\ k>1\qquad\qquad\qquad \mathcal{R}_1^{z} = \left\{\alpha \in \mathcal{R}_1^\infty : \alpha \le \frac{pz}{p-1}\right\}.$$ # The possible rams $r_1 = \dots = r_w$ for totally wildly ramified extensions of degree $p^w$ over $K$ are then just given by $\mathcal{R}_w^{e_K}$, where $e_K$ is the absolute ramification index of $K$. In general, the possible rams $(r_1,\dots,r_w)$ over $K$ are chosen sequentially using the considerations above and the fact that if $r_i < r_{i+1}$ then $L/K$ has an intermediate field $L_i$ with $[L_i:K] = \epsilon p^i$. For example, to get all strictly increasing sequences, each $r_k$ is chosen from $\mathcal{R}_1^{e_K \epsilon p^{k-1}}$. # Ambiguity -- # Let $K$ be be a $p$-adic field # and let $I=(r_1,\dots,r_w)_{\epsilon}^f$ be a # Herbrand invariant which # is compatible with $K$. Let $f_0$ be the residue field degree of $K/\Q_p$, # and set $q=p^{f_0f}$. Let $r_{i_1},\dots,r_{i_k}$ be the # distinct integral rams of $I$, and for # $1\le j\le k$ let $\rho_j$ be the multiplicity of $r_{i_j}$ in $I$. # The **ambiguity** of $I$ over $K$ is # \[\text{Amb}(I/K)=f\cdot\gcd(\epsilon,q-1)\cdot\prod_{j=1}^k\gcd(p^{\rho_j},q).\] # For each $L/K$ in the family # $I/K$, the number of semicanonical polynomials representing $L/K$ is # a divisor of $\frac{\operatorname{Amb}(I/K)}{\lvert\operatorname{Aut}(L/K)\rvert}$. #Field count (field_count) -- # Let $I/K$ be a family of # extensions of $p$-adic fields. The # **field count** of $I/K$ is the number of extensions in $I/K$. #Mass (mass_relative) -- # Let $I/K$ be a family of extensions of # $p$-adic fields with base field $K$. Let $f$ be # the residue field degree of # the family, and let $K_f$ be the unramified # extension of $K$ of degree $f$, which is contained in every $L/K \in I/K$. # To each $L/K\in I/K$ we assign a **mass** $1/|\operatorname{Aut}(L/K_f)|$. # The **mass of the family** $I/K$ is the sum of the # masses of the elements of $I/K$. We also assign an **absolute mass** # $1/|\operatorname{Aut}(L/\Q_p)|$ to each $L/K\in I/K$, and define the # **absolute mass of the family** to be # the sum of the absolute masses of the elements of $I/K$. When # $K = \Q_p$, the mass is the product of the absolute mass with $f$. # For some families, not all of the fields within the family are contained within the LMFDB. In these cases, we display the fraction of the absolute mass among those fields within the database as a percentage.