Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-629563704x-25707687106752\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-629563704xz^2-25707687106752z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-815914561059x-1199405610934208802\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 422142 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-269526786549544680459431933952$ | = | $-1 \cdot 2^{11} \cdot 3^{25} \cdot 7^{4} \cdot 19 \cdot 23^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{186412805526685980326617}{1820685432230184941568} \) | = | $-1 \cdot 2^{-11} \cdot 3^{-25} \cdot 7^{-4} \cdot 13^{6} \cdot 19^{-1} \cdot 23^{-1} \cdot 338017^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.3295132612206393754234125081$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.7617661532560645300200360922$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0584761784362104$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.81164082258063$ | |||
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: | $\Omega$ | ≈ | $0.013131615598691982256854555158$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3131615598691982256854555158 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $25$ = $5^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.313161560 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.013132 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 1.313161560\end{aligned}$$
Modular invariants
Modular form 422142.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 669081600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{11}$ | nonsplit multiplicative | 1 | 1 | 11 | 11 |
| $3$ | $1$ | $I_{25}$ | nonsplit multiplicative | 1 | 1 | 25 | 25 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $23$ | $2$ | $I_{1}^{*}$ | 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 \( 10488 = 2^{3} \cdot 3 \cdot 19 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 2623 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 553 & 2 \\ 553 & 3 \end{array}\right),\left(\begin{array}{rr} 10487 & 2 \\ 10486 & 3 \end{array}\right),\left(\begin{array}{rr} 10033 & 2 \\ 10033 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 10487 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5245 & 2 \\ 5245 & 3 \end{array}\right),\left(\begin{array}{rr} 3497 & 2 \\ 3497 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[10488])$ is a degree-$1212594247434240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10488\Z)$.
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 | $4$ | \( 30153 = 3 \cdot 19 \cdot 23^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 140714 = 2 \cdot 7 \cdot 19 \cdot 23^{2} \) |
| $5$ | good | $2$ | \( 140714 = 2 \cdot 7 \cdot 19 \cdot 23^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 60306 = 2 \cdot 3 \cdot 19 \cdot 23^{2} \) |
| $11$ | good | $2$ | \( 211071 = 3 \cdot 7 \cdot 19 \cdot 23^{2} \) |
| $19$ | 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 422142.n consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 18354.d1, its twist by $-23$.
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 $0$.