Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-22033692x-39718804784\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-22033692xz^2-39718804784z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-352539075x-2542356045250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2791, 6908)$ | $2.2367430110315809756488454895$ | $\infty$ |
| $(-10429/4, 10429/8)$ | $0$ | $2$ |
Integral points
\( \left(-2791, 6908\right) \), \( \left(-2791, -4117\right) \), \( \left(6299, 264038\right) \), \( \left(6299, -270337\right) \), \( \left(20345, 2806364\right) \), \( \left(20345, -2826709\right) \)
Invariants
| Conductor: | $N$ | = | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $2902647157593750000000$ | = | $2^{7} \cdot 3^{7} \cdot 5^{12} \cdot 7^{6} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{882774443450089}{2166000000} \) | = | $2^{-7} \cdot 3^{-1} \cdot 5^{-6} \cdot 19^{-2} \cdot 95929^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9964972395655248859700529205$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.66951706448676320041937426370$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9826442954534543$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.815404573600104$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.2367430110315809756488454895$ |
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| Real period: | $\Omega$ | ≈ | $0.069693418999262207880838277146$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.4941802857839255008351139616 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.494180286 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.069693 \cdot 2.236743 \cdot 64}{2^2} \\ & \approx 2.494180286\end{aligned}$$
Modular invariants
Modular form 418950.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 46448640 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2277 & 4 \\ 2276 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1139 & 0 \end{array}\right),\left(\begin{array}{rr} 457 & 4 \\ 914 & 9 \end{array}\right),\left(\begin{array}{rr} 1522 & 1 \\ 1519 & 0 \end{array}\right),\left(\begin{array}{rr} 1921 & 4 \\ 1562 & 9 \end{array}\right),\left(\begin{array}{rr} 1996 & 289 \\ 1425 & 856 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$363095654400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 11025 = 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 46550 = 2 \cdot 5^{2} \cdot 7^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 16758 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 19 \) |
| $7$ | additive | $26$ | \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 22050 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 418950.u
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 570.a1, its twist by $105$.
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.