Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-89348726x-1119560271244\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-89348726xz^2-1119560271244z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1429579619x-71653286939234\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1770481852/361, 74479767582518/6859)$ | $14.433199074669333799687507372$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462722 \) | = | $2 \cdot 13^{2} \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-495846523052183705180087296$ | = | $-1 \cdot 2^{10} \cdot 13^{10} \cdot 37^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{4652805537}{29246464} \) | = | $-1 \cdot 2^{-10} \cdot 3^{3} \cdot 13^{-4} \cdot 37 \cdot 167^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8059304665463640280039979900$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.11617717938611269706519048853$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9675679702002368$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.290281746991758$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $14.433199074669333799687507372$ |
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| Real period: | $\Omega$ | ≈ | $0.021910905764322427322457614446$ |
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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)$ | ≈ | $2.5299557184222834406220510272 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.529955718 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.021911 \cdot 14.433199 \cdot 8}{1^2} \\ & \approx 2.529955718\end{aligned}$$
Modular invariants
Modular form 462722.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 143216640 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $13$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $37$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 | 4.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 52.16.0-4.b.1.1, level \( 52 = 2^{2} \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right),\left(\begin{array}{rr} 11 & 0 \\ 0 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1 & 2 \end{array}\right),\left(\begin{array}{rr} 49 & 4 \\ 48 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 13 \\ 0 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[52])$ is a degree-$157248$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/52\Z)$.
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 | $4$ | \( 231361 = 13^{2} \cdot 37^{2} \) |
| $5$ | good | $2$ | \( 231361 = 13^{2} \cdot 37^{2} \) |
| $13$ | additive | $98$ | \( 2738 = 2 \cdot 37^{2} \) |
| $37$ | additive | $506$ | \( 338 = 2 \cdot 13^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462722d consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 35594b1, 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.