conductor of an elliptic curve

Let $E$ be an elliptic curve over $\\mathbb{Q}$ . For each prime $p\\in\\mathbb{Z}$ define the quantity $f_{p}$ as follows:

f_{p}=\\begin{cases}0\\text{, if $E$ has good reduction at $p$,}\\\\1\\text{, if $E$ has multiplicative reduction at $p$,}\\\\2\\text{, if $E$ has additive reduction at $p$, and $p\eq 2,3$,}\\\\2+\\delta_{p}\\text{, if $E$ has additive reduction at $p=2\\ or\\ 3$.}\\end{cases}

where $\\delta_{p}$ depends on wild ramification in the action of theinertia group at $p$ of $\\operatorname{Gal}(\\bar{\\mathbb{Q}}/\\mathbb{Q})$ on the Tatemodule $T_{p}(E)$ .

Definition.

The conductor $N_{E/\\mathbb{Q}}$ of ${E/\\mathbb{Q}}$ is defined tobe:

N_{E/\\mathbb{Q}}=\\prod_{p}p^{f_{p}}

where the product is over all primes and the exponent $f_{p}$ isdefined as above.

Example.

Let $E/\\mathbb{Q}\\colon y^{2}+y=x^{3}-x^{2}+2x-2$ . The primes of badreduction for $E$ are $p=5$ and $7$ . The reduction at $p=5$ isadditive, while the reduction at $p=7$ is multiplicative. Hence $N_{E/\\mathbb{Q}}=25\\cdot 7=175$ .

References

1 James Milne, Elliptic Curves, http://www.jmilne.org/math/CourseNotes/math679.htmlonline coursenotes.
2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
3 Joseph H. Silverman, Advanced Topics inthe Arithmetic of Elliptic Curves. Springer-Verlag, New York,1994.