Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-98988708x-214508612032\)
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(homogenize, simplify) |
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\(y^2z=x^3-98988708xz^2-214508612032z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-98988708x-214508612032\)
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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 |
|---|---|---|
| $(-247953491162/29582721, 26116650197998984/160900419519)$ | $22.906635887134686820628804051$ | $\infty$ |
| $(-8606, 0)$ | $0$ | $2$ |
| $(-2288, 0)$ | $0$ | $2$ |
Integral points
\( \left(-8606, 0\right) \), \( \left(-2288, 0\right) \), \( \left(10894, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $42199865145633259584000000$ | = | $2^{12} \cdot 3^{14} \cdot 5^{6} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{7442744143086784}{2927948765625} \) | = | $2^{6} \cdot 3^{-8} \cdot 5^{-6} \cdot 7^{3} \cdot 13^{-4} \cdot 19^{3} \cdot 367^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6155723456174056908608119356$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0906443419926371677192134749$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0447549090132258$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.104458511321262$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $22.906635887134686820628804051$ |
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| Real period: | $\Omega$ | ≈ | $0.049503569098598913448413948770$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.0716818596417407243865158480 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.071681860 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.049504 \cdot 22.906636 \cdot 128}{4^2} \\ & \approx 9.071681860\end{aligned}$$
Modular invariants
Modular form 486720.2.a.ht
For more coefficients, see the Downloads section to the right.
| Modular degree: | 132120576 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | -1 | 6 | 12 | 0 |
| $3$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $13$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
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 | 4.12.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 1039 & 0 \\ 0 & 3119 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 211 & 2082 \\ 2286 & 1039 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 479 & 1032 \\ 876 & 1007 \end{array}\right),\left(\begin{array}{rr} 3113 & 8 \\ 3112 & 9 \end{array}\right),\left(\begin{array}{rr} 779 & 1032 \\ 0 & 1949 \end{array}\right),\left(\begin{array}{rr} 257 & 1038 \\ 522 & 2081 \end{array}\right)$.
The torsion field $K:=\Q(E[3120])$ is a degree-$77290536960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3120\Z)$.
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 | $2$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 10816 = 2^{6} \cdot 13^{2} \) |
| $5$ | nonsplit 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 $d=$
2.
Its isogeny class 486720ht
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 6240z1, 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.