# 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:

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

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

###### Definition.

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

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

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

###### Example.

Let $E/\mathbb{Q}\colon y^{2}+y=x^{3}-x^{2}+2x-2$. The primes of bad reduction for $E$ are $p=5$ and $7$. The reduction at $p=5$ is additive, 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 course notes.
• 2 Joseph H. Silverman, . Springer-Verlag, New York, 1986.
• 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
Title conductor of an elliptic curve ConductorOfAnEllipticCurve 2013-03-22 13:49:51 2013-03-22 13:49:51 alozano (2414) alozano (2414) 9 alozano (2414) Definition msc 14H52 conductor EllipticCurve LSeriesOfAnEllipticCurve ArithmeticOfEllipticCurves conductor of an elliptic curve