Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-47229492x-185067073584\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-47229492xz^2-185067073584z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-755671875x-11845048381250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(671962509/75625, 6375949524048/20796875)$ | $17.256387272668025910027873049$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-8055328920145920000000000$ | = | $-1 \cdot 2^{21} \cdot 3^{3} \cdot 5^{10} \cdot 7^{9} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{1095008203125}{757071872} \) | = | $-1 \cdot 2^{-21} \cdot 3^{3} \cdot 5^{8} \cdot 19^{-2} \cdot 47^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4792958982641331164241736647$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.40401195394387040257904835353$ |
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| $abc$ quality: | $Q$ | ≈ | $1.216195634093843$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.0528142431342395$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $17.256387272668025910027873049$ |
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| Real period: | $\Omega$ | ≈ | $0.027938894835669449952436125653$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8569951140372538672498471285 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.856995114 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.027939 \cdot 17.256387 \cdot 8}{1^2} \\ & \approx 3.856995114\end{aligned}$$
Modular invariants
Modular form 418950.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 112896000 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{21}$ | nonsplit multiplicative | 1 | 1 | 21 | 21 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $19$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 167 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 113 & 2 \\ 113 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 127 & 2 \\ 127 & 3 \end{array}\right),\left(\begin{array}{rr} 85 & 2 \\ 85 & 3 \end{array}\right),\left(\begin{array}{rr} 167 & 2 \\ 166 & 3 \end{array}\right),\left(\begin{array}{rr} 73 & 2 \\ 73 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$74317824$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
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 | $4$ | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 23275 = 5^{2} \cdot 7^{2} \cdot 19 \) |
| $5$ | additive | $2$ | \( 16758 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 19 \) |
| $7$ | additive | $14$ | \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 22050 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 418950.h consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 418950.iu1, its twist by $105$.
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.