Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2565000x-1570826250\)
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(homogenize, simplify) |
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\(y^2z=x^3-2565000xz^2-1570826250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2565000x-1570826250\)
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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: | $\Delta$ | = | $14087689478325000000$ | = | $2^{6} \cdot 3^{9} \cdot 5^{8} \cdot 31^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{3792752640000}{28629151} \) | = | $2^{15} \cdot 3^{3} \cdot 5^{4} \cdot 19^{3} \cdot 31^{-5}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5042102156555680083531415154$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.26071880058511283536425197150$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1070685424336508$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.29595941708384$ | |||
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: | $\Omega$ | ≈ | $0.11935080489918492457911398263$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 10 $ = $ 1\cdot2\cdot1\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.1935080489918492457911398263 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.193508049 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.119351 \cdot 1.000000 \cdot 10}{1^2} \\ & \approx 1.193508049\end{aligned}$$
Modular invariants
Modular form 446400.2.a.mb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10368000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 6 | 6 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $5$ | 5Ns.2.1 | 5.30.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \), index $120$, genus $5$, and generators
$\left(\begin{array}{rr} 3 & 10 \\ 631 & 37 \end{array}\right),\left(\begin{array}{rr} 921 & 10 \\ 920 & 11 \end{array}\right),\left(\begin{array}{rr} 871 & 10 \\ 635 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 5 \\ 15 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 289 \\ 93 & 94 \end{array}\right)$.
The torsion field $K:=\Q(E[930])$ is a degree-$1028505600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/930\Z)$.
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 | $2$ | \( 2325 = 3 \cdot 5^{2} \cdot 31 \) |
| $3$ | additive | $2$ | \( 49600 = 2^{6} \cdot 5^{2} \cdot 31 \) |
| $5$ | additive | $10$ | \( 576 = 2^{6} \cdot 3^{2} \) |
| $31$ | split multiplicative | $32$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 446400mb consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 6975b1, its twist by $120$.
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 $0$.