Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-17326433x+29867839263\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-17326433xz^2+29867839263z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1403441100x+21777865146000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3253, 89100\right) \) | $2.1695835436297563799139049512$ | $\infty$ |
| \( \left(-4847, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([3253:89100:1]\) | $2.1695835436297563799139049512$ | $\infty$ |
| \([-4847:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(29280, 2405700\right) \) | $2.1695835436297563799139049512$ | $\infty$ |
| \( \left(-43620, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-4847, 0\right) \), \((1714,\pm 72171)\), \((3253,\pm 89100)\)
\([-4847:0:1]\), \([1714:\pm 72171:1]\), \([3253:\pm 89100:1]\)
\( \left(-4847, 0\right) \), \((1714,\pm 72171)\), \((3253,\pm 89100)\)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-52635827854891008000000$ | = | $-1 \cdot 2^{17} \cdot 3^{24} \cdot 5^{6} \cdot 7 \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{280880296871140514}{25701087819771} \) | = | $-1 \cdot 2 \cdot 3^{-24} \cdot 7^{-1} \cdot 13^{-1} \cdot 519793^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1025812732256281141088127883$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3159038112153220718006876163$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0089455216720034$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.755719668289419$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.1695835436297563799139049512$ |
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| Real period: | $\Omega$ | ≈ | $0.10970514844604405645648713562$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 2^{2}\cdot( 2^{3} \cdot 3 )\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.424695266559842823244270972 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.424695267 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.109705 \cdot 2.169584 \cdot 192}{2^2} \\ & \approx 11.424695267\end{aligned}$$
Modular invariants
Modular form 436800.2.a.qj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 37748736 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | -1 | 6 | 17 | 0 |
| $3$ | $24$ | $I_{24}$ | split multiplicative | -1 | 1 | 24 | 24 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.12.0.5 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21840 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 14561 & 13120 \\ 2920 & 17601 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21825 & 16 \\ 21824 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 21742 & 21827 \end{array}\right),\left(\begin{array}{rr} 16364 & 3275 \\ 2025 & 2174 \end{array}\right),\left(\begin{array}{rr} 15616 & 5 \\ 9315 & 21826 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 21836 & 21837 \end{array}\right),\left(\begin{array}{rr} 13103 & 0 \\ 0 & 21839 \end{array}\right),\left(\begin{array}{rr} 16136 & 13105 \\ 2095 & 8746 \end{array}\right),\left(\begin{array}{rr} 8189 & 8720 \\ 2730 & 16379 \end{array}\right)$.
The torsion field $K:=\Q(E[21840])$ is a degree-$155817722511360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21840\Z)$.
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 | $4$ | \( 2275 = 5^{2} \cdot 7 \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 145600 = 2^{6} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $14$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 62400 = 2^{6} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 33600 = 2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 436800.qj
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 2184.c3, its twist by $-40$.
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.