Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-436697x+108551036\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-436697xz^2+108551036z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6987155x+6940279150\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(328, 558)$ | $2.8537784179535479069485848924$ | $\infty$ |
| $(1864, 75018)$ | $4.7903258210306308089866157948$ | $\infty$ |
| $(1331/4, -1331/8)$ | $0$ | $2$ |
Integral points
\( \left(328, 558\right) \), \( \left(328, -886\right) \), \( \left(1864, 75018\right) \), \( \left(1864, -76882\right) \), \( \left(22444, 3349688\right) \), \( \left(22444, -3372132\right) \)
Invariants
| Conductor: | $N$ | = | \( 442225 \) | = | $5^{2} \cdot 7^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $249762401026326125$ | = | $5^{3} \cdot 7^{6} \cdot 19^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{13312053}{361} \) | = | $3^{3} \cdot 19^{-2} \cdot 79^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1185163094436834194383750382$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.72901773277571855676900488277$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8861406626336358$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8904804583362966$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.131746529719904639329361489$ |
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| Real period: | $\Omega$ | ≈ | $0.31071810326921359402167234982$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $15.078213084229731903596660761 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 15.078213084 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.310718 \cdot 12.131747 \cdot 16}{2^2} \\ & \approx 15.078213084\end{aligned}$$
Modular invariants
Modular form 442225.2.a.cg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4147200 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 380 = 2^{2} \cdot 5 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 377 & 4 \\ 376 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 286 \\ 284 & 95 \end{array}\right),\left(\begin{array}{rr} 308 & 1 \\ 303 & 0 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 42 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[380])$ is a degree-$472780800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/380\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 88445 = 5 \cdot 7^{2} \cdot 19^{2} \) |
| $5$ | additive | $10$ | \( 17689 = 7^{2} \cdot 19^{2} \) |
| $7$ | additive | $26$ | \( 9025 = 5^{2} \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 442225cg
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 475c2, its twist by $133$.
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.