Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-14132035713x-646620089769951\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-14132035713xz^2-646620089769951z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1144694892780x-471389479526972592\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-8303738256436066444271619160762972914157144752473545/120790020484003917486128995027008164086830899041, 1725863012428711142115408291666189971282298196722153518925793737983509708192/41980400970426365290844782509653404321794691878768675816788211409068911)$ | $115.69168675648646996132658754$ | $\infty$ |
| $(137269, 0)$ | $0$ | $2$ |
Integral points
\( \left(137269, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 485184 \) | = | $2^{6} \cdot 3 \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $2367076573238321347005775872$ | = | $2^{54} \cdot 3 \cdot 7^{2} \cdot 19^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{25309080274342544331625}{191933498523648} \) | = | $2^{-36} \cdot 3^{-1} \cdot 5^{3} \cdot 7^{-2} \cdot 19^{-1} \cdot 43^{3} \cdot 136559^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.4275999061430975455298391809$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9156596457199593513994772828$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0385639997055625$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.242511739217657$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $115.69168675648646996132658754$ |
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| Real period: | $\Omega$ | ≈ | $0.013846772941216027825519011660$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.4078260708134301280549468797 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.407826071 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.013847 \cdot 115.691687 \cdot 16}{2^2} \\ & \approx 6.407826071\end{aligned}$$
Modular invariants
Modular form 485184.2.a.ce
For more coefficients, see the Downloads section to the right.
| Modular degree: | 716636160 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{44}^{*}$ | additive | -1 | 6 | 54 | 36 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 | 2.3.0.1 |
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 2395 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7499 & 9540 \\ 8898 & 9311 \end{array}\right),\left(\begin{array}{rr} 4787 & 0 \\ 0 & 9575 \end{array}\right),\left(\begin{array}{rr} 5540 & 9567 \\ 6473 & 9398 \end{array}\right),\left(\begin{array}{rr} 9541 & 36 \\ 9540 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 2692 & 27 \\ 877 & 3134 \end{array}\right),\left(\begin{array}{rr} 9557 & 9540 \\ 6336 & 8225 \end{array}\right),\left(\begin{array}{rr} 4823 & 9540 \\ 4824 & 9539 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[9576])$ is a degree-$1715626967040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9576\Z)$.
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 | $2$ | \( 1083 = 3 \cdot 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 161728 = 2^{6} \cdot 7 \cdot 19^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 69312 = 2^{6} \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 485184.ce
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 798.d2, its twist by $152$.
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.