Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-250493633x-1018353655137\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-250493633xz^2-1018353655137z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-20289984300x-742318944642000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(61206817/1369, 444921750000/50653)$ | $12.837950019921380008371346185$ | $\infty$ |
| $(-4407, 0)$ | $0$ | $2$ |
| $(17563, 0)$ | $0$ | $2$ |
Integral points
\( \left(-13157, 0\right) \), \( \left(-4407, 0\right) \), \( \left(17563, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $558005687156736000000000000$ | = | $2^{30} \cdot 3^{2} \cdot 5^{12} \cdot 7^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{424378956393532177129}{136231857216000000} \) | = | $2^{-12} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-2} \cdot 13^{-6} \cdot 43^{3} \cdot 174763^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8357930179204959272708395680$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9913532908635277758446117192$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0086784378946845$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.3614536316367705$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.837950019921380008371346185$ |
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| Real period: | $\Omega$ | ≈ | $0.038913920988292801709217431898$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 384 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.989799345444941951832790910 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.989799345 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.038914 \cdot 12.837950 \cdot 384}{4^2} \\ & \approx 11.989799345\end{aligned}$$
Modular invariants
Modular form 436800.2.a.nj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 191102976 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{20}^{*}$ | additive | 1 | 6 | 30 | 12 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $6$ | $I_{6}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
| $3$ | 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 $384$, genus $5$, and generators
$\left(\begin{array}{rr} 9 & 4 \\ 10904 & 10913 \end{array}\right),\left(\begin{array}{rr} 5471 & 10908 \\ 5472 & 10907 \end{array}\right),\left(\begin{array}{rr} 10911 & 10916 \\ 5476 & 5467 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 4201 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10909 & 12 \\ 10908 & 13 \end{array}\right),\left(\begin{array}{rr} 5459 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 3277 & 12 \\ 3282 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8579 & 10908 \\ 7794 & 10847 \end{array}\right),\left(\begin{array}{rr} 7273 & 10914 \\ 7286 & 5465 \end{array}\right),\left(\begin{array}{rr} 8183 & 10914 \\ 5508 & 5501 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$4869303828480$ 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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | 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.nj
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2730.o4, 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.