Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+28290x+9154462\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+28290xz^2+9154462z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+36664461x+427000597326\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-94, 2427)$ | $0.30312919730336536200136367270$ | $\infty$ |
Integral points
\( \left(-94, 2427\right) \), \( \left(-94, -2334\right) \), \( \left(-88, 2490\right) \), \( \left(-88, -2403\right) \), \( \left(320, 6981\right) \), \( \left(320, -7302\right) \), \( \left(647402, 520584450\right) \), \( \left(647402, -521231853\right) \)
Invariants
| Conductor: | $N$ | = | \( 422142 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-37633869338584806$ | = | $-1 \cdot 2 \cdot 3^{7} \cdot 7 \cdot 19^{2} \cdot 23^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{16915218263}{254221254} \) | = | $2^{-1} \cdot 3^{-7} \cdot 7^{-1} \cdot 17^{3} \cdot 19^{-2} \cdot 23^{-1} \cdot 151^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8611971168971665989488272444$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.29345000893259175354545082850$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8633211368547657$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.519014356324497$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.30312919730336536200136367270$ |
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| Real period: | $\Omega$ | ≈ | $0.27095906914564033351635650138$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 56 $ = $ 1\cdot7\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.5995938874023614811185223864 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.599593887 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.270959 \cdot 0.303129 \cdot 56}{1^2} \\ & \approx 4.599593887\end{aligned}$$
Modular invariants
Modular form 422142.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4730880 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $23$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2761 & 2 \\ 2761 & 3 \end{array}\right),\left(\begin{array}{rr} 1933 & 2 \\ 1933 & 3 \end{array}\right),\left(\begin{array}{rr} 3863 & 2 \\ 3862 & 3 \end{array}\right),\left(\begin{array}{rr} 1289 & 2 \\ 1289 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 3863 & 0 \end{array}\right),\left(\begin{array}{rr} 967 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2857 & 2 \\ 2857 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[3864])$ is a degree-$19855344402432$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3864\Z)$.
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 | $4$ | \( 11109 = 3 \cdot 7 \cdot 23^{2} \) |
| $3$ | split multiplicative | $4$ | \( 140714 = 2 \cdot 7 \cdot 19 \cdot 23^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 20102 = 2 \cdot 19 \cdot 23^{2} \) |
| $19$ | nonsplit multiplicative | $20$ | \( 22218 = 2 \cdot 3 \cdot 7 \cdot 23^{2} \) |
| $23$ | additive | $288$ | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 422142u consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 18354o1, its twist by $-23$.
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.