Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-81909633x-290987791137\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-81909633xz^2-290987791137z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6634680300x-212110195698000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(11723, 600000)$ | $3.1163635540703763102941086493$ | $\infty$ |
| $(10473, 0)$ | $0$ | $2$ |
Integral points
\( \left(10473, 0\right) \), \((11723,\pm 600000)\), \((208203,\pm 94910400)\)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-1401241959936000000000000$ | = | $-1 \cdot 2^{21} \cdot 3^{4} \cdot 5^{12} \cdot 7 \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{14837772556740428569}{342100087875000} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-4} \cdot 5^{-6} \cdot 7^{-1} \cdot 13^{-6} \cdot 19^{3} \cdot 283^{3} \cdot 457^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4206427572512000770736373959$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5762030301942319256474095471$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9880308093217611$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.106250414827283$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1163635540703763102941086493$ |
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| Real period: | $\Omega$ | ≈ | $0.025057838618374470608219306453$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 384 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot1\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.4965761613516217427269068779 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.496576161 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.025058 \cdot 3.116364 \cdot 384}{2^2} \\ & \approx 7.496576161\end{aligned}$$
Modular invariants
Modular form 436800.2.a.kn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 95551488 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | 1 | 6 | 21 | 3 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2183 & 10908 \\ 2178 & 10847 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 10870 & 10911 \end{array}\right),\left(\begin{array}{rr} 1570 & 3 \\ 7773 & 10912 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 10909 & 12 \\ 10908 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 3195 & 5008 \\ 3158 & 4997 \end{array}\right),\left(\begin{array}{rr} 10910 & 10917 \\ 5487 & 8 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4201 & 12 \\ 3366 & 73 \end{array}\right),\left(\begin{array}{rr} 3641 & 12 \\ 9100 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$19477215313920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 175 = 5^{2} \cdot 7 \) |
| $3$ | split multiplicative | $4$ | \( 11200 = 2^{6} \cdot 5^{2} \cdot 7 \) |
| $5$ | additive | $18$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | nonsplit 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, 3 and 6.
Its isogeny class 436800kn
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2730o4, 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.