Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-37443x+3016258\)
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(homogenize, simplify) |
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\(y^2z=x^3-37443xz^2+3016258z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-37443x+3016258\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(71, 846)$ | $3.2204599836390671913776036612$ | $\infty$ |
Integral points
\((71,\pm 846)\)
Invariants
| Conductor: | $N$ | = | \( 458640 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-570621542400000$ | = | $-1 \cdot 2^{17} \cdot 3^{7} \cdot 5^{5} \cdot 7^{2} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{39678209809}{3900000} \) | = | $-1 \cdot 2^{-5} \cdot 3^{-1} \cdot 5^{-5} \cdot 7 \cdot 13^{-1} \cdot 1783^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5722468704125613627711430362$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.0054751873426756568053961723734$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8873627433757527$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.326337160281496$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.2204599836390671913776036612$ |
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| Real period: | $\Omega$ | ≈ | $0.50498923212279692245055933277$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.5051904568803507241449168253 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.505190457 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.504989 \cdot 3.220460 \cdot 4}{1^2} \\ & \approx 6.505190457\end{aligned}$$
Modular invariants
Modular form 458640.2.a.f
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1843200 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{9}^{*}$ | additive | -1 | 4 | 17 | 5 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 521 & 2 \\ 521 & 3 \end{array}\right),\left(\begin{array}{rr} 781 & 2 \\ 781 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1559 & 0 \end{array}\right),\left(\begin{array}{rr} 1559 & 2 \\ 1558 & 3 \end{array}\right),\left(\begin{array}{rr} 937 & 2 \\ 937 & 3 \end{array}\right),\left(\begin{array}{rr} 391 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1081 & 2 \\ 1081 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$463743221760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 28665 = 3^{2} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 50960 = 2^{4} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 91728 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $14$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 35280 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 458640.f consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 19110.bw1, its twist by $12$.
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.