Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-29213633x+17396200863\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-29213633xz^2+17396200863z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2366304300x+12688929342000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-677, 192000)$ | $2.0239695237645865734528948472$ | $\infty$ |
| $(603, 0)$ | $0$ | $2$ |
Integral points
\((-677,\pm 192000)\), \( \left(603, 0\right) \), \((22573,\pm 3295500)\)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $1464764928786432000000000$ | = | $2^{24} \cdot 3^{3} \cdot 5^{9} \cdot 7^{3} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{673163386034885929}{357608625192000} \) | = | $2^{-6} \cdot 3^{-3} \cdot 5^{-3} \cdot 7^{-3} \cdot 13^{-6} \cdot 29^{3} \cdot 47^{3} \cdot 643^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3283954796776256371708736782$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4839557526206574857446458294$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0044067734582287$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.865089612667182$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.0239695237645865734528948472$ |
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| Real period: | $\Omega$ | ≈ | $0.074532683231753011296440709024$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 288 $ = $ 2^{2}\cdot3\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)$ | ≈ | $10.861335315753690931911848255 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.861335316 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.074533 \cdot 2.023970 \cdot 288}{2^2} \\ & \approx 10.861335316\end{aligned}$$
Modular invariants
Modular form 436800.2.a.oz
For more coefficients, see the Downloads section to the right.
| Modular degree: | 71663616 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{14}^{*}$ | additive | 1 | 6 | 24 | 6 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $6$ | $I_{6}$ | 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 |
| $3$ | 3Cs | 3.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 32760 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 16389 \\ 16227 & 28432 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 32725 & 36 \\ 32724 & 37 \end{array}\right),\left(\begin{array}{rr} 16379 & 32724 \\ 32742 & 32111 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 12313 & 15018 \\ 17082 & 19819 \end{array}\right),\left(\begin{array}{rr} 18742 & 3 \\ 2421 & 32452 \end{array}\right),\left(\begin{array}{rr} 32750 & 32757 \\ 7059 & 152 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right),\left(\begin{array}{rr} 17665 & 36 \\ 4074 & 31369 \end{array}\right)$.
The torsion field $K:=\Q(E[32760])$ is a degree-$175294937825280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/32760\Z)$.
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 | $2$ | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| $3$ | split multiplicative | $4$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | 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 436800oz
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 2730p3, 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.