Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-10500x-114048\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-10500xz^2-114048z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-850527x-85692546\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(864, 25200)$ | $3.7978739125301541254120057678$ | $\infty$ |
| $(-96, 0)$ | $0$ | $2$ |
| $(-11, 0)$ | $0$ | $2$ |
Integral points
\( \left(-96, 0\right) \), \( \left(-11, 0\right) \), \( \left(108, 0\right) \), \((864,\pm 25200)\)
Invariants
| Conductor: | $N$ | = | \( 485520 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $68125874745600$ | = | $2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{20720464}{11025} \) | = | $2^{4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 109^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3457456439188027880248089778$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.53295914848260212504477974544$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8898550693671632$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.008566623681773$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.7978739125301541254120057678$ |
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| Real period: | $\Omega$ | ≈ | $0.50109248880973898007853826279$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.6123443640612634165008825417 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.612344364 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.501092 \cdot 3.797874 \cdot 64}{4^2} \\ & \approx 7.612344364\end{aligned}$$
Modular invariants
Modular form 485520.2.a.dh
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1310720 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 4 | 8 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $17$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7140 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 171 & 5882 \\ 2686 & 1259 \end{array}\right),\left(\begin{array}{rr} 2381 & 5882 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3569 & 1258 \\ 0 & 7139 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 5879 & 0 \\ 0 & 7139 \end{array}\right),\left(\begin{array}{rr} 6121 & 4624 \\ 3842 & 2109 \end{array}\right),\left(\begin{array}{rr} 7137 & 4 \\ 7136 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[7140])$ is a degree-$7277201326080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7140\Z)$.
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 | $2$ | \( 289 = 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 161840 = 2^{4} \cdot 5 \cdot 7 \cdot 17^{2} \) |
| $5$ | split multiplicative | $6$ | \( 97104 = 2^{4} \cdot 3 \cdot 7 \cdot 17^{2} \) |
| $7$ | split multiplicative | $8$ | \( 69360 = 2^{4} \cdot 3 \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 485520dh
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 840e2, its twist by $-68$.
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.