Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+2748560x+11142216788\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+2748560xz^2+11142216788z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+222633333x+8122008138426\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-454, 99000)$ | $0.47065868295362219961342138138$ | $\infty$ |
| $(-1829, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1829, 0\right) \), \((-454,\pm 99000)\), \((1196,\pm 127050)\), \((2396,\pm 177450)\), \((7466,\pm 669240)\), \((18026,\pm 2432760)\)
Invariants
| Conductor: | $N$ | = | \( 446160 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-54952455898454400000000$ | = | $-1 \cdot 2^{13} \cdot 3^{5} \cdot 5^{8} \cdot 11^{4} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{116149984977671}{2779502343750} \) | = | $2^{-1} \cdot 3^{-5} \cdot 5^{-8} \cdot 11^{-4} \cdot 97^{3} \cdot 503^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0436056102280897896686673927$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0679837509373761122246915505$ |
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| $abc$ quality: | $Q$ | ≈ | $1.042494685403093$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.596325750903382$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.47065868295362219961342138138$ |
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| Real period: | $\Omega$ | ≈ | $0.083858702518368293491098054730$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1280 $ = $ 2^{2}\cdot5\cdot2^{3}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.630024474078343090369062477 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.630024474 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.083859 \cdot 0.470659 \cdot 1280}{2^2} \\ & \approx 12.630024474\end{aligned}$$
Modular invariants
Modular form 446160.2.a.id
For more coefficients, see the Downloads section to the right.
| Modular degree: | 35389440 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $11$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $2$ | $I_0^{*}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 937 & 2912 \\ 52 & 1353 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3426 & 3427 \end{array}\right),\left(\begin{array}{rr} 791 & 0 \\ 0 & 3431 \end{array}\right),\left(\begin{array}{rr} 3425 & 8 \\ 3424 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3197 & 1352 \\ 2782 & 1611 \end{array}\right),\left(\begin{array}{rr} 92 & 793 \\ 3367 & 534 \end{array}\right),\left(\begin{array}{rr} 1288 & 2483 \\ 169 & 1782 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$531372441600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 | $4$ | \( 507 = 3 \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 148720 = 2^{4} \cdot 5 \cdot 11 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 29744 = 2^{4} \cdot 11 \cdot 13^{2} \) |
| $11$ | split 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 and 4.
Its isogeny class 446160id
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 330a4, 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.