Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-8923256x-10262639100\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-8923256xz^2-10262639100z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-722783763x-7479295552638\)
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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 |
|---|---|---|
| $(4359, 183534)$ | $6.1484656115311966035808580981$ | $\infty$ |
| $(6739, 485484)$ | $8.4942690802037803970594872749$ | $\infty$ |
| $(-1725, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1725, 0\right) \), \((4359,\pm 183534)\), \((6739,\pm 485484)\)
Invariants
| Conductor: | $N$ | = | \( 446160 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $12233064729600$ | = | $2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{15897679904620804}{2475} \) | = | $2^{2} \cdot 3^{-2} \cdot 5^{-2} \cdot 11^{-1} \cdot 23^{3} \cdot 71^{3} \cdot 97^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3578628095068925949524274694$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.49776548030950313574465698073$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0052432979893915$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.583651077233076$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $51.226041464178644546673681295$ |
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| Real period: | $\Omega$ | ≈ | $0.087351167197848596457011267846$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $17.898618051285574813466157209 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 17.898618051 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.087351 \cdot 51.226041 \cdot 16}{2^2} \\ & \approx 17.898618051\end{aligned}$$
Modular invariants
Modular form 446160.2.a.fn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9437184 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_0^{*}$ | 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 | 16.24.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11440 = 2^{4} \cdot 5 \cdot 11 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 974 & 3289 \\ 8437 & 4666 \end{array}\right),\left(\begin{array}{rr} 8799 & 0 \\ 0 & 11439 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 11436 & 11437 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 11342 & 11427 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 2653 & 3536 \\ 9724 & 105 \end{array}\right),\left(\begin{array}{rr} 11425 & 16 \\ 11424 & 17 \end{array}\right),\left(\begin{array}{rr} 4941 & 3536 \\ 7488 & 10245 \end{array}\right),\left(\begin{array}{rr} 9608 & 8801 \\ 1599 & 10570 \end{array}\right)$.
The torsion field $K:=\Q(E[11440])$ is a degree-$21254897664000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11440\Z)$.
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 | $2$ | \( 1859 = 11 \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 148720 = 2^{4} \cdot 5 \cdot 11 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 89232 = 2^{4} \cdot 3 \cdot 11 \cdot 13^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40560 = 2^{4} \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 446160fn
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1320h4, 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.