Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+73008x+9846954\)
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(homogenize, simplify) |
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\(y^2z=x^3+73008xz^2+9846954z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+73008x+9846954\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-65, 2197)$ | $1.3471415216889329092580957267$ | $\infty$ |
| $(331305/1156, 289654677/39304)$ | $7.1698170294347234400959458866$ | $\infty$ |
Integral points
\((-65,\pm 2197)\)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-66793075570802880$ | = | $-1 \cdot 2^{6} \cdot 3^{9} \cdot 5 \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{7077888}{10985} \) | = | $2^{18} \cdot 3^{3} \cdot 5^{-1} \cdot 13^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9136397998862170674643851138$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.53936768562560622381740859540$ |
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| $abc$ quality: | $Q$ | ≈ | $1.001928731766074$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4919388747136315$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.3725166288079147708064027737$ |
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| Real period: | $\Omega$ | ≈ | $0.23682747579867770356520707907$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $17.757355640653686183266446619 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 17.757355641 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.236827 \cdot 9.372517 \cdot 8}{1^2} \\ & \approx 17.757355641\end{aligned}$$
Modular invariants
Modular form 486720.2.a.nq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3483648 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | 1 | 6 | 6 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 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 \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 391 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1494 & 1229 \\ 1427 & 1476 \end{array}\right),\left(\begin{array}{rr} 157 & 6 \\ 81 & 19 \end{array}\right),\left(\begin{array}{rr} 1555 & 6 \\ 1554 & 7 \end{array}\right),\left(\begin{array}{rr} 779 & 0 \\ 0 & 1559 \end{array}\right),\left(\begin{array}{rr} 301 & 6 \\ 123 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 385 & 6 \\ 384 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$57967902720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $2$ | \( 2535 = 3 \cdot 5 \cdot 13^{2} \) |
| $3$ | additive | $2$ | \( 54080 = 2^{6} \cdot 5 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 97344 = 2^{6} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 486720nq
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 585b1, its twist by $-312$.
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.