Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-187064793x+3029993290058\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-187064793xz^2+3029993290058z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-242435971107x+141368094248871006\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(102691254659854728/9716393454769, 45103800591016397282465105/30287096350975361897)$ | $37.533792166492924312720955847$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 436810 \) | = | $2 \cdot 5 \cdot 11^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-3547227881158627844527734550$ | = | $-1 \cdot 2 \cdot 5^{2} \cdot 11^{15} \cdot 19^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{24061071481921}{117897384550} \) | = | $-1 \cdot 2^{-1} \cdot 5^{-2} \cdot 11^{-9} \cdot 19 \cdot 31^{3} \cdot 349^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9708109760536851010833391938$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.80890402021020618904634911689$ |
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| $abc$ quality: | $Q$ | ≈ | $0.95651699415333$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.467077264580851$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $37.533792166492924312720955847$ |
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| Real period: | $\Omega$ | ≈ | $0.038559815139302043218993834672$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.7891843496678011954282071288 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.789184350 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.038560 \cdot 37.533792 \cdot 4}{1^2} \\ & \approx 5.789184350\end{aligned}$$
Modular invariants
Modular form 436810.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 390044160 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{9}^{*}$ | additive | -1 | 2 | 15 | 9 |
| $19$ | $1$ | $IV^{*}$ | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 199 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 259 & 6 \\ 258 & 7 \end{array}\right),\left(\begin{array}{rr} 261 & 262 \\ 254 & 257 \end{array}\right),\left(\begin{array}{rr} 119 & 258 \\ 93 & 245 \end{array}\right),\left(\begin{array}{rr} 133 & 6 \\ 135 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 146 & 123 \\ 175 & 139 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$60825600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $4$ | \( 43681 = 11^{2} \cdot 19^{2} \) |
| $5$ | split multiplicative | $6$ | \( 87362 = 2 \cdot 11^{2} \cdot 19^{2} \) |
| $11$ | additive | $72$ | \( 3610 = 2 \cdot 5 \cdot 19^{2} \) |
| $19$ | additive | $146$ | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 436810.g
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 39710.n1, its twist by $209$.
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.