Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-52413x+20694917\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-52413xz^2+20694917z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-838603x+1323636102\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(63, 4168)$ | $1.4805645443911646458408672764$ | $\infty$ |
| $(-337, 168)$ | $0$ | $2$ |
Integral points
\( \left(-337, 168\right) \), \( \left(63, 4168\right) \), \( \left(63, -4232\right) \), \( \left(385, 7388\right) \), \( \left(385, -7774\right) \)
Invariants
| Conductor: | $N$ | = | \( 429590 \) | = | $2 \cdot 5 \cdot 7 \cdot 17 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-175567700551040000$ | = | $-1 \cdot 2^{10} \cdot 5^{4} \cdot 7^{3} \cdot 17 \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{338463151209}{3731840000} \) | = | $-1 \cdot 2^{-10} \cdot 3^{3} \cdot 5^{-4} \cdot 7^{-3} \cdot 17^{-1} \cdot 23^{3} \cdot 101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9909245673218140485469864038$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.51870507773859381854247268786$ |
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| $abc$ quality: | $Q$ | ≈ | $1.041224641716976$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6399800446677664$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4805645443911646458408672764$ |
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| Real period: | $\Omega$ | ≈ | $0.27321717731036188762992233492$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 120 $ = $ ( 2 \cdot 5 )\cdot2\cdot3\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.135469969330679851020730467 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.135469969 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.273217 \cdot 1.480565 \cdot 120}{2^2} \\ & \approx 12.135469969\end{aligned}$$
Modular invariants
Modular form 429590.2.a.bw
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3144960 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $17$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 952 = 2^{3} \cdot 7 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 121 & 834 \\ 832 & 119 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 949 & 4 \\ 948 & 5 \end{array}\right),\left(\begin{array}{rr} 618 & 1 \\ 167 & 0 \end{array}\right),\left(\begin{array}{rr} 818 & 1 \\ 543 & 0 \end{array}\right),\left(\begin{array}{rr} 477 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[952])$ is a degree-$20214448128$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/952\Z)$.
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 | $4$ | \( 42959 = 7 \cdot 17 \cdot 19^{2} \) |
| $3$ | good | $2$ | \( 61370 = 2 \cdot 5 \cdot 17 \cdot 19^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 42959 = 7 \cdot 17 \cdot 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 61370 = 2 \cdot 5 \cdot 17 \cdot 19^{2} \) |
| $17$ | split multiplicative | $18$ | \( 25270 = 2 \cdot 5 \cdot 7 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1190 = 2 \cdot 5 \cdot 7 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 429590bw
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1190a1, its twist by $-19$.
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.