Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-10108120x+12129113840\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-10108120xz^2+12129113840z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-818757747x+8844580262574\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2039, 0)$ | $0$ | $2$ |
Integral points
\( \left(2039, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 476520 \) | = | $2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $2509267768794627287040$ | = | $2^{11} \cdot 3^{16} \cdot 5 \cdot 11^{2} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1185450336504002}{26043266205} \) | = | $2 \cdot 3^{-16} \cdot 5^{-1} \cdot 11^{-2} \cdot 167^{3} \cdot 503^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8941812106393171116388135253$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.78657680554281368133517036468$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9967634929262502$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.589179744252089$ | |||
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: | $\Omega$ | ≈ | $0.14450856346196563399702748584$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 1\cdot2^{4}\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.3121370153914501439524397735 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.312137015 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.144509 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 2.312137015\end{aligned}$$
Modular invariants
Modular form 476520.2.a.cj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 28311552 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | 1 | 3 | 11 | 0 |
| $3$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $2$ | $I_0^{*}$ | 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 | 16.24.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 16720 = 2^{4} \cdot 5 \cdot 11 \cdot 19 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 16493 & 8816 \\ 15504 & 1445 \end{array}\right),\left(\begin{array}{rr} 2639 & 0 \\ 0 & 16719 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 16705 & 16 \\ 16704 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 16622 & 16707 \end{array}\right),\left(\begin{array}{rr} 2832 & 13205 \\ 14915 & 10546 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 16716 & 16717 \end{array}\right),\left(\begin{array}{rr} 10356 & 6821 \\ 10279 & 15410 \end{array}\right),\left(\begin{array}{rr} 10451 & 8816 \\ 14630 & 4181 \end{array}\right)$.
The torsion field $K:=\Q(E[16720])$ is a degree-$99851304960000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/16720\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 1805 = 5 \cdot 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 158840 = 2^{3} \cdot 5 \cdot 11 \cdot 19^{2} \) |
| $5$ | split multiplicative | $6$ | \( 95304 = 2^{3} \cdot 3 \cdot 11 \cdot 19^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 43320 = 2^{3} \cdot 3 \cdot 5 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 476520cj
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1320h5, its twist by $-19$.
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 $0$.