Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-1005347256x+12269702674800\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-1005347256xz^2+12269702674800z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-81433127763x+8944368950545938\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(978562/49, 135776850/343)$ | $8.8858579297816939270776208852$ | $\infty$ |
| $(18300, 0)$ | $0$ | $2$ |
| $(18313, 0)$ | $0$ | $2$ |
Integral points
\( \left(-36612, 0\right) \), \( \left(18300, 0\right) \), \( \left(18313, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 446160 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $24596954746644234240000$ | = | $2^{18} \cdot 3^{2} \cdot 5^{4} \cdot 11^{2} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{5683972151443376419201}{1244117160000} \) | = | $2^{-6} \cdot 3^{-2} \cdot 5^{-4} \cdot 11^{-2} \cdot 13^{-4} \cdot 17846401^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6800092106431711812799342098$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7043873513524575038359583676$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9996929468325244$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.67319677010109$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.8858579297816939270776208852$ |
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| Real period: | $\Omega$ | ≈ | $0.094912643201015300735119246943$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.7470421057942591014447807473 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.747042106 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.094913 \cdot 8.885858 \cdot 128}{4^2} \\ & \approx 6.747042106\end{aligned}$$
Modular invariants
Modular form 446160.2.a.be
For more coefficients, see the Downloads section to the right.
| Modular degree: | 123863040 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
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 |
|---|---|---|
| $2$ | 2Cs | 8.24.0.15 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17160 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 17156 & 17157 \end{array}\right),\left(\begin{array}{rr} 17159 & 12866 \\ 0 & 4289 \end{array}\right),\left(\begin{array}{rr} 17153 & 8 \\ 17152 & 9 \end{array}\right),\left(\begin{array}{rr} 3127 & 6 \\ 4674 & 17155 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 10297 & 8 \\ 6868 & 33 \end{array}\right),\left(\begin{array}{rr} 3 & 12874 \\ 12866 & 17155 \end{array}\right),\left(\begin{array}{rr} 11447 & 6 \\ 17154 & 17155 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 14519 & 17152 \\ 6596 & 17127 \end{array}\right)$.
The torsion field $K:=\Q(E[17160])$ is a degree-$63764692992000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17160\Z)$.
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 | $2$ | \( 169 = 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 148720 = 2^{4} \cdot 5 \cdot 11 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 89232 = 2^{4} \cdot 3 \cdot 11 \cdot 13^{2} \) |
| $11$ | split multiplicative | $12$ | \( 40560 = 2^{4} \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 446160be
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 4290bc3, its twist by $-52$.
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.