Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+5796096x+2654805084\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+5796096xz^2+2654805084z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+469483749x+1936761357510\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-443, 0)$ | $0$ | $2$ |
Integral points
\( \left(-443, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 424536 \) | = | $2^{3} \cdot 3 \cdot 7^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-15511132259783297872896$ | = | $-1 \cdot 2^{10} \cdot 3 \cdot 7^{7} \cdot 19^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{3799448348}{2736741} \) | = | $2^{2} \cdot 3^{-1} \cdot 7^{-1} \cdot 19^{-4} \cdot 983^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9463672850658817939560639742$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.076429929511616179782152881347$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9002780392111306$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.5013353065028125$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.078983895943751851051684223410$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.2637423351000296168269475746 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.263742335 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.078984 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 1.263742335\end{aligned}$$
Modular invariants
Modular form 424536.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 35389440 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 3 | 10 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $19$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3185 & 8 \\ 3184 & 9 \end{array}\right),\left(\begin{array}{rr} 2732 & 3191 \\ 1345 & 3186 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 403 & 402 \\ 1210 & 2803 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1200 & 2801 \\ 1225 & 1272 \end{array}\right),\left(\begin{array}{rr} 2183 & 3184 \\ 2348 & 3159 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3186 & 3187 \end{array}\right),\left(\begin{array}{rr} 1072 & 3 \\ 1069 & 2 \end{array}\right)$.
The torsion field $K:=\Q(E[3192])$ is a degree-$381250437120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3192\Z)$.
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 | $2$ | \( 53067 = 3 \cdot 7^{2} \cdot 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 141512 = 2^{3} \cdot 7^{2} \cdot 19^{2} \) |
| $7$ | additive | $32$ | \( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 424536n
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3192a4, its twist by $133$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.