Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-11335227x-12771634815\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-11335227xz^2-12771634815z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-14690454219x-595829322566010\)
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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 |
|---|---|---|
| $(-2220, 39225)$ | $2.4042427690841237258866708403$ | $\infty$ |
| $(-1338, 669)$ | $0$ | $2$ |
| $(3830, -1915)$ | $0$ | $2$ |
Integral points
\( \left(-2220, 39225\right) \), \( \left(-2220, -37005\right) \), \( \left(-1338, 669\right) \), \( \left(3830, -1915\right) \), \( \left(24312, 3740439\right) \), \( \left(24312, -3764751\right) \)
Invariants
| Conductor: | $N$ | = | \( 405042 \) | = | $2 \cdot 3 \cdot 11 \cdot 17 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $22756637383068625500816$ | = | $2^{4} \cdot 3^{10} \cdot 11^{6} \cdot 17^{2} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{3423676911662954233}{483711578981136} \) | = | $2^{-4} \cdot 3^{-10} \cdot 7^{3} \cdot 11^{-6} \cdot 17^{-2} \cdot 139^{3} \cdot 1549^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0159118884439373952019408114$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5436923988607171651974270955$ |
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| $abc$ quality: | $Q$ | ≈ | $1.070078751332218$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.67356470884948$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.4042427690841237258866708403$ |
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| Real period: | $\Omega$ | ≈ | $0.083053059086053708876322866804$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1920 $ = $ 2^{2}\cdot( 2 \cdot 5 )\cdot( 2 \cdot 3 )\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $23.961566010955333330576203247 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 23.961566011 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.083053 \cdot 2.404243 \cdot 1920}{4^2} \\ & \approx 23.961566011\end{aligned}$$
Modular invariants
Modular form 405042.2.a.dp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 53084160 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $17$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 42636 = 2^{2} \cdot 3 \cdot 11 \cdot 17 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 27855 & 40394 \\ 24814 & 2243 \end{array}\right),\left(\begin{array}{rr} 21319 & 38152 \\ 40394 & 33669 \end{array}\right),\left(\begin{array}{rr} 11629 & 38152 \\ 21014 & 33669 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 20195 & 0 \\ 0 & 42635 \end{array}\right),\left(\begin{array}{rr} 42633 & 4 \\ 42632 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 14213 & 40394 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[42636])$ is a degree-$12221799727104000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/42636\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 361 = 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 12274 = 2 \cdot 17 \cdot 19^{2} \) |
| $5$ | good | $2$ | \( 135014 = 2 \cdot 11 \cdot 17 \cdot 19^{2} \) |
| $11$ | split multiplicative | $12$ | \( 36822 = 2 \cdot 3 \cdot 17 \cdot 19^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 23826 = 2 \cdot 3 \cdot 11 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 405042.dp
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1122.c2, its twist by $-19$.
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.