Tate's algorithm

Assume that all the coefficients of the equation of the curve lie in a complete discrete valuation ring R with perfect residue field K and maximal ideal generated by a prime π. The elliptic curve is given by the equation

${\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.}$

Define:

${\displaystyle v(\Delta )=}$ the p-adic valuation of ${\displaystyle \pi }$ in ${\displaystyle \Delta }$, that is, exponent of ${\displaystyle \pi }$ in prime factorization of ${\displaystyle \Delta }$, or infinity if ${\displaystyle \Delta =0}$

${\displaystyle a_{i,m}=a_{i}/\pi ^{m}}$

${\displaystyle b_{2}=a_{1}^{2}+4a_{2}}$

${\displaystyle b_{4}=a_{1}a_{3}+2a_{4}^{}}$

${\displaystyle b_{6}=a_{3}^{2}+4a_{6}}$

${\displaystyle b_{8}=a_{1}^{2}a_{6}-a_{1}a_{3}a_{4}+4a_{2}a_{6}+a_{2}a_{3}^{2}-a_{4}^{2}}$

${\displaystyle c_{4}=b_{2}^{2}-24b_{4}}$

${\displaystyle c_{6}=-b_{2}^{3}+36b_{2}b_{4}-216b_{6}}$

${\displaystyle \Delta =-b_{2}^{2}b_{8}-8b_{4}^{3}-27b_{6}^{2}+9b_{2}b_{4}b_{6}}$

${\displaystyle j=c_{4}^{3}/\Delta .}$

• Step 1: If π does not divide Δ then the type is I₀, c=1 and f=0.
• Step 2: If π divides Δ but not c₄ then the type is I_v with v = v(Δ), c=v, and f=1.
• Step 3. Otherwise, change coordinates so that π divides a₃,a₄,a₆. If π² does not divide a₆ then the type is II, c=1, and f=v(Δ);
• Step 4. Otherwise, if π³ does not divide b₈ then the type is III, c=2, and f=v(Δ)−1;
• Step 5. Otherwise, let Q₁ be the polynomial

${\displaystyle Q_{1}(Y)=Y^{2}+a_{3,1}Y-a_{6,2}.}$.

If π³ does not divide b₆ then the type is IV, c=3 if ${\displaystyle Q_{1}(Y)}$ has two roots in K and 1 if it has two roots outside of K, and f=v(Δ)−2.

• Step 6. Otherwise, change coordinates so that π divides a₁ and a₂, π² divides a₃ and a₄, and π³ divides a₆. Let P be the polynomial

${\displaystyle P(T)=T^{3}+a_{2,1}T^{2}+a_{4,2}T+a_{6,3}.}$

If ${\displaystyle P(T)}$ has 3 distinct roots modulo π then the type is I₀^*, f=v(Δ)−4, and c is 1+(number of roots of P in K).

• Step 7. If P has one single and one double root, then the type is I_ν^* for some ν>0, f=v(Δ)−4−ν, c=2 or 4: there is a "sub-algorithm" for dealing with this case.
• Step 8. If P has a triple root, change variables so the triple root is 0, so that π² divides a₂ and π³ divides a₄, and π⁴ divides a₆. Let Q₂ be the polynomial

${\displaystyle Q_{2}(Y)=Y^{2}+a_{3,2}Y-a_{6,4}.}$.

If ${\displaystyle Q_{2}(Y)}$ has two distinct roots modulo π then the type is IV^*, f=v(Δ)−6, and c is 3 if the roots are in K, 1 otherwise.

• Step 9. If ${\displaystyle Q_{2}(Y)}$ has a double root, change variables so the double root is 0. Then π³ divides a₃ and π⁵ divides a₆. If π⁴ does not divide a₄ then the type is III^* and f=v(Δ)−7 and c = 2.
• Step 10. Otherwise if π⁶ does not divide a₆ then the type is II^* and f=v(Δ)−8 and c = 1.
• Step 11. Otherwise the equation is not minimal. Divide each a_n by πⁿ and go back to step 1.

The algorithm is implemented for algebraic number fields in the PARI/GP computer algebra system, available through the function elllocalred.

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