The **Faltings height** of an elliptic curve $E$ defined over $\Q$ is the quantity
$$
-\frac{1}{2}\log(A),
$$
where $A$ is the covolume (that is, the area of a fundamental period parallelogram) of the Néron lattice of $E$.

The **stable Faltings height** of $E$ is
$$
\frac{1}{12}(\log\mathrm{denom}(j)-\log(|\Delta|)) - \frac{1}{2}\log(A),
$$
where $j$ is the $j$-invariant of $E$, $\Delta$ the discriminant of any model of $E$ and $A$ the covolume of the period lattice of that model.
The stable height is independent of the model of $E$, and the unstable and stable heights are equal for semistable curves, for which $\mathrm{denom}(j)=|\Delta|$.

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**Knowl status:**

- Review status: beta
- Last edited by John Cremona on 2021-03-22 09:53:05

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