Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-881810x+374237650\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-881810xz^2+374237650z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-14108963x+23937100638\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(993, 21344)$ | $0.71155543938656931962013414885$ | $\infty$ |
| $(14545, 1743295)$ | $6.3967772754837661469006029724$ | $\infty$ |
Integral points
\( \left(271, 12319\right) \), \( \left(271, -12590\right) \), \( \left(993, 21344\right) \), \( \left(993, -22337\right) \), \( \left(14545, 1743295\right) \), \( \left(14545, -1757840\right) \)
Invariants
| Conductor: | $N$ | = | \( 436810 \) | = | $2 \cdot 5 \cdot 11^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-16548079858462350550$ | = | $-1 \cdot 2 \cdot 5^{2} \cdot 11^{7} \cdot 19^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{2520369}{550} \) | = | $-1 \cdot 2^{-1} \cdot 3^{3} \cdot 5^{-2} \cdot 11^{-1} \cdot 17^{3} \cdot 19$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4085392888184455150423396487$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.75336766702503339699465042821$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7996097480733245$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.081137695184443$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.4909466236546193507601121364$ |
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| Real period: | $\Omega$ | ≈ | $0.21013860130836702152474579924$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 1\cdot2\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $11.324654904543780620630960741 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.324654905 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.210139 \cdot 4.490947 \cdot 12}{1^2} \\ & \approx 11.324654905\end{aligned}$$
Modular invariants
Modular form 436810.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9192960 |
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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 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 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $19$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
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 \( 88 = 2^{3} \cdot 11 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 87 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 87 & 2 \\ 86 & 3 \end{array}\right),\left(\begin{array}{rr} 23 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 2 \\ 45 & 3 \end{array}\right),\left(\begin{array}{rr} 57 & 2 \\ 57 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[88])$ is a degree-$10137600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/88\Z)$.
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 | $4$ | \( 43681 = 11^{2} \cdot 19^{2} \) |
| $5$ | nonsplit 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 no rational isogenies. Its isogeny class 436810q consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 39710c1, 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.