# 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:

$${f}_{p}=\{\begin{array}{cc}0\text{, if}E\text{has good reduction at}p\text{,}\hfill & \\ 1\text{, if}E\text{has multiplicative reduction at}p\text{,}\hfill & \\ 2\text{, if}E\text{has additive reduction at}p\text{, and}p\ne 2,3\text{,}\hfill & \\ 2+{\delta}_{p}\text{, if}E\text{has additive reduction at}p=2or\mathrm{\hspace{0.25em}3}\text{.}\hfill & \end{array}$$ |

where ${\delta}_{p}$ depends on wild ramification in the action of the inertia group at $p$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the Tate module ${T}_{p}(E)$.

###### Definition.

The conductor^{} ${N}_{E\mathrm{/}\mathrm{Q}}$ of $E\mathrm{/}\mathrm{Q}$ 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 $E/\mathbb{Q}:{y}^{2}+y={x}^{3}-{x}^{2}+2x-2$. 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, The Arithmetic of Elliptic Curves. 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 |
---|---|

Canonical name | ConductorOfAnEllipticCurve |

Date of creation | 2013-03-22 13:49:51 |

Last modified on | 2013-03-22 13:49:51 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 9 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 14H52 |

Synonym | conductor |

Related topic | EllipticCurve |

Related topic | LSeriesOfAnEllipticCurve |

Related topic | ArithmeticOfEllipticCurves |

Defines | conductor of an elliptic curve |