Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+23004852x+17756899472\)
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(homogenize, simplify) |
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\(y^2z=x^3+23004852xz^2+17756899472z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+23004852x+17756899472\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(816604, 737946288)$ | $8.8028071989763769058979956184$ | $\infty$ |
Integral points
\((816604,\pm 737946288)\)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-915393742771322880000000$ | = | $-1 \cdot 2^{29} \cdot 3^{17} \cdot 5^{7} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{41689615345255319}{28343520000000} \) | = | $2^{-11} \cdot 3^{-11} \cdot 5^{-7} \cdot 13 \cdot 31^{3} \cdot 67^{3} \cdot 71^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2864611272704017015727948902$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2699426525195061024070761826$ |
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| $abc$ quality: | $Q$ | ≈ | $1.063662624594901$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.770151266590722$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.8028071989763769058979956184$ |
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| Real period: | $\Omega$ | ≈ | $0.055731410155081550661604003894$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9247428681780562579961603543 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.924742868 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.055731 \cdot 8.802807 \cdot 8}{1^2} \\ & \approx 3.924742868\end{aligned}$$
Modular invariants
Modular form 486720.2.a.bw
For more coefficients, see the Downloads section to the right.
| Modular degree: | 62447616 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{19}^{*}$ | additive | -1 | 6 | 29 | 11 |
| $3$ | $4$ | $I_{11}^{*}$ | additive | -1 | 2 | 17 | 11 |
| $5$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $13$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
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 \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 31 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 119 & 2 \\ 118 & 3 \end{array}\right),\left(\begin{array}{rr} 61 & 2 \\ 61 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 119 & 0 \end{array}\right),\left(\begin{array}{rr} 97 & 2 \\ 97 & 3 \end{array}\right),\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$17694720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $4$ | \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 54080 = 2^{6} \cdot 5 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 97344 = 2^{6} \cdot 3^{2} \cdot 13^{2} \) |
| $7$ | good | $2$ | \( 97344 = 2^{6} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $38$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 486720bw consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 5070n1, its twist by $24$.
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.