Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-532746252x-4694626426896\)
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(homogenize, simplify) |
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\(y^2z=x^3-532746252xz^2-4694626426896z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-532746252x-4694626426896\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(757658382/11449, 19364313536640/1225043)$ | $15.377374807491078385887286384$ | $\infty$ |
| $(26628, 0)$ | $0$ | $2$ |
Integral points
\( \left(26628, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 411840 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $155938380716100377640960000$ | = | $2^{23} \cdot 3^{6} \cdot 5^{4} \cdot 11^{12} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{87501897507774086005761}{815991377947460000} \) | = | $2^{-5} \cdot 3^{3} \cdot 5^{-4} \cdot 11^{-12} \cdot 13^{-1} \cdot 3229^{3} \cdot 4583^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8472299910930672844212366591$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2582030759190944745977658585$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0322408901773137$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.5609607447525145$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $15.377374807491078385887286384$ |
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| Real period: | $\Omega$ | ≈ | $0.031442270506839908267956249588$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.7359932541151959364958689581 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.735993254 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.031442 \cdot 15.377375 \cdot 64}{2^2} \\ & \approx 7.735993254\end{aligned}$$
Modular invariants
Modular form 411840.2.a.jo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 165150720 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{13}^{*}$ | additive | 1 | 6 | 23 | 5 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $13$ | $1$ | $I_{1}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17160 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 15019 & 3576 \\ 738 & 9301 \end{array}\right),\left(\begin{array}{rr} 9688 & 3 \\ 7485 & 5722 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 17154 & 17155 \end{array}\right),\left(\begin{array}{rr} 17153 & 8 \\ 17152 & 9 \end{array}\right),\left(\begin{array}{rr} 10297 & 11448 \\ 12588 & 11473 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 2144 & 16437 \\ 711 & 12122 \end{array}\right),\left(\begin{array}{rr} 5719 & 0 \\ 0 & 17159 \end{array}\right),\left(\begin{array}{rr} 7801 & 11448 \\ 2604 & 11473 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[17160])$ is a degree-$255058771968000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17160\Z)$.
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 | $4$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 4160 = 2^{6} \cdot 5 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 82368 = 2^{6} \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 37440 = 2^{6} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 31680 = 2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 411840.jo
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1430.h2, its twist by $-24$.
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.