Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+19839417x+24193487032\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+19839417xz^2+24193487032z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+19839417x+24193487032\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(43327108466668/19036548729, 748159693555811091074/2626529737786317)$ | $29.859848236796704427864172800$ | $\infty$ |
| $(-1144, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1144, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-752626302255140625000000$ | = | $-1 \cdot 2^{6} \cdot 3^{10} \cdot 5^{12} \cdot 13^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{3834800837445824}{3342041015625} \) | = | $2^{6} \cdot 3^{-4} \cdot 5^{-12} \cdot 13^{-2} \cdot 109^{3} \cdot 359^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2689987553374330361521958749$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0906443419926371677192134749$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1541079700404029$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.7362384575400025$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $29.859848236796704427864172800$ |
|
| Real period: | $\Omega$ | ≈ | $0.058475944109989661797151281602$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.9843312665107897455514837785 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.984331267 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.058476 \cdot 29.859848 \cdot 16}{2^2} \\ & \approx 6.984331267\end{aligned}$$
Modular invariants
Modular form 486720.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 66060288 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
|
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 |
| $3$ | $2$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 |
|---|---|---|
| $2$ | 2B | 4.12.0.9 |
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
$\left(\begin{array}{rr} 2497 & 1056 \\ 1668 & 1105 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 64 & 205 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1985 & 1914 \\ 2406 & 1133 \end{array}\right),\left(\begin{array}{rr} 2069 & 114 \\ 318 & 557 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 4 & 49 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1039 & 0 \\ 0 & 3119 \end{array}\right),\left(\begin{array}{rr} 704 & 309 \\ 2205 & 2504 \end{array}\right),\left(\begin{array}{rr} 3105 & 16 \\ 3104 & 17 \end{array}\right)$.
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 and 4.
Its isogeny class 486720s
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 6240z4, 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.