Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+1802487x-431606469\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+1802487xz^2-431606469z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2336023125x-20172071756250\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 476850 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-455552299806914062500$ | = | $-1 \cdot 2^{2} \cdot 3 \cdot 5^{10} \cdot 11^{5} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{2747555975}{1932612} \) | = | $2^{-2} \cdot 3^{-1} \cdot 5^{2} \cdot 11^{-5} \cdot 479^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6524970169956402145585544076$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.10530791539421813773351234569$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9936024056521444$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.193332011451356$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.094059266639315764282168291287$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.7029633319657882141084145643 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $25$ = $5^2$ (exact) |
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BSD formula
$$\begin{aligned} 4.702963332 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.094059 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 4.702963332\end{aligned}$$
Modular invariants
Modular form 476850.2.a.gd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 26400000 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $11$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $17$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11220 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 7481 & 5950 \\ 1105 & 7311 \end{array}\right),\left(\begin{array}{rr} 7919 & 0 \\ 0 & 11219 \end{array}\right),\left(\begin{array}{rr} 5611 & 5950 \\ 2975 & 7311 \end{array}\right),\left(\begin{array}{rr} 7141 & 5950 \\ 10625 & 7311 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 11165 & 11101 \end{array}\right),\left(\begin{array}{rr} 11211 & 10 \\ 11210 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8907 & 5270 \\ 10540 & 6731 \end{array}\right)$.
The torsion field $K:=\Q(E[11220])$ is a degree-$47648342016000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11220\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 238425 = 3 \cdot 5^{2} \cdot 11 \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 158950 = 2 \cdot 5^{2} \cdot 11 \cdot 17^{2} \) |
| $5$ | additive | $2$ | \( 1734 = 2 \cdot 3 \cdot 17^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 43350 = 2 \cdot 3 \cdot 5^{2} \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 476850gd
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 1650d1, its twist by $85$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.