Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-384238x-48672469\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-384238xz^2-48672469z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-497972475x-2263393118250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(809, 12663)$ | $0.99599221046196451402238032367$ | $\infty$ |
| $(-379, 6723)$ | $1.3627014945422515419371344353$ | $\infty$ |
Integral points
\( \left(-541, 1377\right) \), \( \left(-541, -837\right) \), \( \left(-401, 6613\right) \), \( \left(-401, -6213\right) \), \( \left(-379, 6723\right) \), \( \left(-379, -6345\right) \), \( \left(-159, 2983\right) \), \( \left(-159, -2825\right) \), \( \left(769, 10167\right) \), \( \left(769, -10937\right) \), \( \left(809, 12663\right) \), \( \left(809, -13473\right) \), \( \left(2041, 86583\right) \), \( \left(2041, -88625\right) \), \( \left(347111, 204330843\right) \), \( \left(347111, -204677955\right) \)
Invariants
| Conductor: | $N$ | = | \( 417450 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $2613931169256000000$ | = | $2^{9} \cdot 3^{6} \cdot 5^{6} \cdot 11^{7} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{226646274673}{94431744} \) | = | $2^{-9} \cdot 3^{-6} \cdot 7^{3} \cdot 11^{-1} \cdot 13^{3} \cdot 23^{-1} \cdot 67^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2310897684240256788546547899$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.22742317580779021952330333430$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9181480536650909$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8781459994658807$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2883172244113019035536467220$ |
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| Real period: | $\Omega$ | ≈ | $0.19896977287364704976506211457$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 3^{2}\cdot2\cdot1\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $18.456205358183336369401136276 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 18.456205358 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.198970 \cdot 1.288317 \cdot 72}{1^2} \\ & \approx 18.456205358\end{aligned}$$
Modular invariants
Modular form 417450.2.a.fd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11197440 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $11$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $23$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 30360 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 23 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 7591 & 12150 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12143 & 0 \\ 0 & 30359 \end{array}\right),\left(\begin{array}{rr} 3961 & 12150 \\ 17955 & 6091 \end{array}\right),\left(\begin{array}{rr} 30355 & 6 \\ 30354 & 7 \end{array}\right),\left(\begin{array}{rr} 15181 & 12150 \\ 21255 & 6091 \end{array}\right),\left(\begin{array}{rr} 16446 & 26065 \\ 21505 & 12651 \end{array}\right),\left(\begin{array}{rr} 2759 & 18210 \\ 2205 & 24269 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[30360])$ is a degree-$7800313872384000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/30360\Z)$.
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 | $4$ | \( 69575 = 5^{2} \cdot 11^{2} \cdot 23 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 69575 = 5^{2} \cdot 11^{2} \cdot 23 \) |
| $5$ | additive | $14$ | \( 16698 = 2 \cdot 3 \cdot 11^{2} \cdot 23 \) |
| $11$ | additive | $72$ | \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 18150 = 2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 417450fd
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 1518p1, its twist by $-55$.
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.