Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+2908500x-4480670000\)
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(homogenize, simplify) |
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\(y^2z=x^3+2908500xz^2-4480670000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2908500x-4480670000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2474, 133632)$ | $3.4939790331755020174472860812$ | $\infty$ |
| $(17834, 2391552)$ | $4.5587170270141572759944381575$ | $\infty$ |
Integral points
\((1301,\pm 38799)\), \((2474,\pm 133632)\), \((17834,\pm 2391552)\)
Invariants
| Conductor: | $N$ | = | \( 446400 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $-10247667764428800000000$ | = | $-1 \cdot 2^{27} \cdot 3^{8} \cdot 5^{8} \cdot 31^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{36450495095}{137276928} \) | = | $2^{-9} \cdot 3^{-2} \cdot 5 \cdot 7^{3} \cdot 31^{-3} \cdot 277^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9062626135885525202680961043$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.24427709012517946071078574817$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9552998756360596$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.45609601023047$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $15.125094170260136042726648313$ |
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| Real period: | $\Omega$ | ≈ | $0.065340851876611097477543195318$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $15.812584604780185287207359880 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 15.812584605 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.065341 \cdot 15.125094 \cdot 16}{1^2} \\ & \approx 15.812584605\end{aligned}$$
Modular invariants
Modular form 446400.2.a.mo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 19906560 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{17}^{*}$ | additive | -1 | 6 | 27 | 9 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $31$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 744 = 2^{3} \cdot 3 \cdot 31 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 185 & 738 \\ 0 & 743 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 741 & 742 \\ 734 & 737 \end{array}\right),\left(\begin{array}{rr} 214 & 525 \\ 641 & 365 \end{array}\right),\left(\begin{array}{rr} 371 & 738 \\ 369 & 725 \end{array}\right),\left(\begin{array}{rr} 739 & 6 \\ 738 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 313 & 6 \\ 195 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[744])$ is a degree-$4114022400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/744\Z)$.
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 | $8$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 17856 = 2^{6} \cdot 3^{2} \cdot 31 \) |
| $31$ | nonsplit multiplicative | $32$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 446400mo
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 4650ba2, 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.