Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1384233708x+19816551254032\)
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(homogenize, simplify) |
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\(y^2z=x^3-1384233708xz^2+19816551254032z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1384233708x+19816551254032\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-135902/9, 164845304/27)$ | $7.4649620591991761069660432001$ | $\infty$ |
| $(21788, 0)$ | $0$ | $2$ |
Integral points
\( \left(21788, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $104851432982911172972544000$ | = | $2^{15} \cdot 3^{22} \cdot 5^{3} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{2543984126301795848}{909361981125} \) | = | $2^{3} \cdot 3^{-16} \cdot 5^{-3} \cdot 11^{6} \cdot 13^{-2} \cdot 5641^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9621459358973783455694279963$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2639311371326234950735215052$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0847562556631796$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.7087670353475$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.4649620591991761069660432001$ |
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| Real period: | $\Omega$ | ≈ | $0.058475944109989661797151281602$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.9843312665107897455514837785 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.984331267 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.058476 \cdot 7.464962 \cdot 64}{2^2} \\ & \approx 6.984331267\end{aligned}$$
Modular invariants
Modular form 486720.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 264241152 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 6 | 15 | 0 |
| $3$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 | 8.12.0.10 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 1039 & 0 \\ 0 & 3119 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 166 \\ 2834 & 3075 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 8 & 65 \end{array}\right),\left(\begin{array}{rr} 1286 & 2337 \\ 3027 & 2060 \end{array}\right),\left(\begin{array}{rr} 479 & 2064 \\ 1752 & 911 \end{array}\right),\left(\begin{array}{rr} 1678 & 3 \\ 1341 & 1060 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 2333 & 2850 \\ 768 & 2267 \end{array}\right),\left(\begin{array}{rr} 3105 & 16 \\ 3104 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[3120])$ is a degree-$77290536960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3120\Z)$.
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 | $8$ | \( 10816 = 2^{6} \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 97344 = 2^{6} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 486720s
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 6240z2, its twist by $-312$.
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.