Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+7850718x-17999735436\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+7850718xz^2-17999735436z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+10174529853x-839948274453186\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(13169012/1369, 49068488478/50653)$ | $11.169721379201363225122288177$ | $\infty$ |
| $(1684, -842)$ | $0$ | $2$ |
Integral points
\( \left(1684, -842\right) \)
Invariants
| Conductor: | $N$ | = | \( 479370 \) | = | $2 \cdot 3 \cdot 5 \cdot 19 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-170982395701110374400000$ | = | $-1 \cdot 2^{20} \cdot 3^{5} \cdot 5^{5} \cdot 19^{2} \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{89962967236397039}{287450726400000} \) | = | $2^{-20} \cdot 3^{-5} \cdot 5^{-5} \cdot 19^{-2} \cdot 29^{3} \cdot 15451^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1411844895082521533928939845$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4575365745150151398012579683$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0281324592856305$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.644472150327595$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $11.169721379201363225122288177$ |
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| Real period: | $\Omega$ | ≈ | $0.051995275023140911953684079553$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2\cdot1\cdot5\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.8077273504343170024840173224 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.807727350 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.051995 \cdot 11.169721 \cdot 40}{2^2} \\ & \approx 5.807727350\end{aligned}$$
Modular invariants
Modular form 479370.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 60480000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{20}$ | nonsplit multiplicative | 1 | 1 | 20 | 20 |
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $19$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $29$ | $2$ | $I_0^{*}$ | 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 | 2.3.0.1 |
| $5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 33060 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \cdot 29 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 12181 & 1160 \\ 6670 & 11601 \end{array}\right),\left(\begin{array}{rr} 18239 & 0 \\ 0 & 33059 \end{array}\right),\left(\begin{array}{rr} 20156 & 25085 \\ 5655 & 23926 \end{array}\right),\left(\begin{array}{rr} 21896 & 25085 \\ 19575 & 25202 \end{array}\right),\left(\begin{array}{rr} 33041 & 20 \\ 33040 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 16531 & 1160 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 32820 & 32711 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[33060])$ is a degree-$644948656128000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/33060\Z)$.
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 | $4$ | \( 12615 = 3 \cdot 5 \cdot 29^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 159790 = 2 \cdot 5 \cdot 19 \cdot 29^{2} \) |
| $5$ | split multiplicative | $6$ | \( 15979 = 19 \cdot 29^{2} \) |
| $19$ | split multiplicative | $20$ | \( 25230 = 2 \cdot 3 \cdot 5 \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 479370.l
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 570.l4, its twist by $29$.
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.