Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-6639633x-4833277137\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-6639633xz^2-4833277137z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-537810300x-3521845602000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4263, 210600)$ | $5.0968373331453400599097846587$ | $\infty$ |
| $(-807, 0)$ | $0$ | $2$ |
Integral points
\( \left(-807, 0\right) \), \((4263,\pm 210600)\)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $8650687500000000000000$ | = | $2^{14} \cdot 3^{2} \cdot 5^{18} \cdot 7 \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{126449185587012304}{33791748046875} \) | = | $2^{4} \cdot 3^{-2} \cdot 5^{-12} \cdot 7^{-1} \cdot 13^{-3} \cdot 17^{3} \cdot 11717^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9176473480554720614184731162$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3042566811851523464646559746$ |
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| $abc$ quality: | $Q$ | ≈ | $1.025107934414267$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.522850657794275$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.0968373331453400599097846587$ |
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| Real period: | $\Omega$ | ≈ | $0.095927183948989519560330928407$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.734206057953045125990835803 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.734206058 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.095927 \cdot 5.096837 \cdot 96}{2^2} \\ & \approx 11.734206058\end{aligned}$$
Modular invariants
Modular form 436800.2.a.na
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23887872 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | 1 | 6 | 14 | 0 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{12}^{*}$ | additive | 1 | 2 | 18 | 12 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 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 |
|---|---|---|
| $2$ | 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
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 7066 & 6555 \\ 2325 & 7636 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1819 & 6540 \\ 10550 & 6479 \end{array}\right),\left(\begin{array}{rr} 9839 & 6540 \\ 9840 & 6539 \end{array}\right),\left(\begin{array}{rr} 2183 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 6074 & 4365 \\ 2055 & 3284 \end{array}\right),\left(\begin{array}{rr} 7734 & 1535 \\ 7735 & 5914 \end{array}\right),\left(\begin{array}{rr} 10909 & 12 \\ 10908 & 13 \end{array}\right),\left(\begin{array}{rr} 10909 & 6550 \\ 5510 & 5469 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 5459 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 10870 & 10911 \end{array}\right)$.
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 | $2$ | \( 2275 = 5^{2} \cdot 7 \cdot 13 \) |
| $3$ | 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 436800.na
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 5460.e1, 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.