Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-77100x+17662000\)
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(homogenize, simplify) |
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\(y^2z=x^3-77100xz^2+17662000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-77100x+17662000\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 446400 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $-105428680704000000$ | = | $-1 \cdot 2^{23} \cdot 3^{3} \cdot 5^{6} \cdot 31^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{458314011}{953312} \) | = | $-1 \cdot 2^{-5} \cdot 3^{3} \cdot 31^{-3} \cdot 257^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9555201332621923609842696947$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.16357266596180321329076946333$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9685438880978434$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6049737253618472$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.29790178184437068215317618323$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot2\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.5748213821324481858381141988 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.574821382 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.297902 \cdot 1.000000 \cdot 12}{1^2} \\ & \approx 3.574821382\end{aligned}$$
Modular invariants
Modular form 446400.2.a.rq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4976640 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{13}^{*}$ | additive | 1 | 6 | 23 | 5 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $31$ | $3$ | $I_{3}$ | split 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 \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3715 & 6 \\ 3714 & 7 \end{array}\right),\left(\begin{array}{rr} 2791 & 750 \\ 3165 & 2251 \end{array}\right),\left(\begin{array}{rr} 1801 & 750 \\ 195 & 2251 \end{array}\right),\left(\begin{array}{rr} 1859 & 2970 \\ 3345 & 1469 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1086 & 3385 \\ 3475 & 2406 \end{array}\right),\left(\begin{array}{rr} 743 & 0 \\ 0 & 3719 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3720])$ is a degree-$1974730752000$ 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 | $4$ | \( 2325 = 3 \cdot 5^{2} \cdot 31 \) |
| $3$ | additive | $6$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 17856 = 2^{6} \cdot 3^{2} \cdot 31 \) |
| $31$ | split multiplicative | $32$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 446400rq
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 558b1, its twist by $40$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.