Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+164268x+81622944\)
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(homogenize, simplify) |
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\(y^2z=x^3+164268xz^2+81622944z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+164268x+81622944\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-182, 6760)$ | $1.0619647629780131169577758128$ | $\infty$ |
| $(-42, 8640)$ | $2.6388442165969695490736721176$ | $\infty$ |
| $(-312, 0)$ | $0$ | $2$ |
Integral points
\( \left(-312, 0\right) \), \((-182,\pm 6760)\), \((-42,\pm 8640)\), \((120,\pm 10152)\), \((325,\pm 13013)\), \((1378,\pm 54080)\), \((3198,\pm 182520)\), \((12324,\pm 1368900)\), \((13378,\pm 1548080)\)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-3161802393884160000$ | = | $-1 \cdot 2^{12} \cdot 3^{9} \cdot 5^{4} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{1259712}{8125} \) | = | $2^{6} \cdot 3^{9} \cdot 5^{-4} \cdot 13^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2315835119381527245571492263$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.56799756385364322143326054363$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9255042656096455$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8149440639405765$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.7928770868116557314663237309$ |
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| Real period: | $\Omega$ | ≈ | $0.18299861382054306094458240851$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $16.354964334646064151880380860 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 16.354964335 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.182999 \cdot 2.792877 \cdot 128}{2^2} \\ & \approx 16.354964335\end{aligned}$$
Modular invariants
Modular form 486720.2.a.kk
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6193152 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 6 | 12 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 156 = 2^{2} \cdot 3 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 134 & 1 \\ 11 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 153 & 4 \\ 152 & 5 \end{array}\right),\left(\begin{array}{rr} 56 & 1 \\ 103 & 0 \end{array}\right),\left(\begin{array}{rr} 121 & 40 \\ 116 & 39 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[156])$ is a degree-$10063872$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/156\Z)$.
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 | $2$ | \( 507 = 3 \cdot 13^{2} \) |
| $3$ | additive | $2$ | \( 54080 = 2^{6} \cdot 5 \cdot 13^{2} \) |
| $5$ | split 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.
Its isogeny class 486720kk
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 18720w1, 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.