Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-20088300x+34671886000\)
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(homogenize, simplify) |
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\(y^2z=x^3-20088300xz^2+34671886000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-20088300x+34671886000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2501, 8649)$ | $1.3827598186338598434659509766$ | $\infty$ |
Integral points
\((2501,\pm 8649)\)
Invariants
| Conductor: | $N$ | = | \( 446400 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $-512917120917504000000$ | = | $-1 \cdot 2^{19} \cdot 3^{7} \cdot 5^{6} \cdot 31^{5} $ |
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| j-invariant: | $j$ | = | \( -\frac{300238092661681}{171774906} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 29^{3} \cdot 31^{-5} \cdot 2309^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9200861814841390411630514582$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52634031009311604403920099094$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0100188200478981$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.770672254080682$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3827598186338598434659509766$ |
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| Real period: | $\Omega$ | ≈ | $0.16313851524192521116278235992$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2\cdot2^{2}\cdot1\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.0232553499248673876629950492 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.023255350 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.163139 \cdot 1.382760 \cdot 40}{1^2} \\ & \approx 9.023255350\end{aligned}$$
Modular invariants
Modular form 446400.2.a.nz
For more coefficients, see the Downloads section to the right.
| Modular degree: | 21504000 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{9}^{*}$ | additive | 1 | 6 | 19 | 1 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $31$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 2783 & 1850 \\ 40 & 1359 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 3665 & 3601 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 3714 & 3715 \\ 1865 & 4 \end{array}\right),\left(\begin{array}{rr} 1866 & 5 \\ 925 & 3716 \end{array}\right),\left(\begin{array}{rr} 3711 & 10 \\ 3710 & 11 \end{array}\right),\left(\begin{array}{rr} 3714 & 3715 \\ 2485 & 4 \end{array}\right),\left(\begin{array}{rr} 3366 & 5 \\ 2155 & 3716 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3720])$ is a degree-$658243584000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3720\Z)$.
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 | $4$ | \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \) |
| $3$ | additive | $8$ | \( 49600 = 2^{6} \cdot 5^{2} \cdot 31 \) |
| $5$ | additive | $14$ | \( 576 = 2^{6} \cdot 3^{2} \) |
| $31$ | split multiplicative | $32$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 446400.nz
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 186.c1, its twist by $-120$.
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.