Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-6598608x-16048732396\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-6598608xz^2-16048732396z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8551796643x-748641381721506\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(265533262754/13039321, 135129791217860032/47084988131)$ | $23.478205239068839212399763448$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462722 \) | = | $2 \cdot 13^{2} \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-92840473304183301255364$ | = | $-1 \cdot 2^{2} \cdot 13^{6} \cdot 37^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{1369}{4} \) | = | $-1 \cdot 2^{-2} \cdot 37^{2}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0940513905633073955549992657$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1975215487043146761118241809$ |
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| $abc$ quality: | $Q$ | ≈ | $0.910040393716602$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.639413034146393$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $23.478205239068839212399763448$ |
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| Real period: | $\Omega$ | ≈ | $0.043585632604953467889139378049$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.1864994221899850221318469096 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.186499422 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.043586 \cdot 23.478205 \cdot 8}{1^2} \\ & \approx 8.186499422\end{aligned}$$
Modular invariants
Modular form 462722.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 40919040 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $37$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1924 = 2^{2} \cdot 13 \cdot 37 \), index $4$, genus $0$, and generators
$\left(\begin{array}{rr} 1663 & 0 \\ 0 & 1923 \end{array}\right),\left(\begin{array}{rr} 1 & 481 \\ 0 & 963 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 739 & 0 \\ 0 & 1923 \end{array}\right),\left(\begin{array}{rr} 1921 & 4 \\ 1920 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 1919 & 1917 \end{array}\right)$.
The torsion field $K:=\Q(E[1924])$ is a degree-$1146134126592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1924\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 231361 = 13^{2} \cdot 37^{2} \) |
| $13$ | additive | $86$ | \( 2738 = 2 \cdot 37^{2} \) |
| $37$ | additive | $290$ | \( 338 = 2 \cdot 13^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462722.h consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 2738.b1, its twist by $481$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.