Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-8019435227x-276414394165621\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-8019435227xz^2-276414394165621z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-128310963627x-17690649537563354\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 402930 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-2355002173215252000000000$ | = | $-1 \cdot 2^{11} \cdot 3^{8} \cdot 5^{9} \cdot 11^{6} \cdot 37^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{44164307457093068844199489}{1823508000000000} \) | = | $-1 \cdot 2^{-11} \cdot 3^{-2} \cdot 5^{-9} \cdot 7^{3} \cdot 23^{3} \cdot 37^{-3} \cdot 79^{3} \cdot 27791^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1629879929411980908479171250$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4147342122079579731193227176$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0503294865642252$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.200665726731417$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.0079768901600587794536647089164$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 594 $ = $ 11\cdot2\cdot3^{2}\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.7382727550749149954768370963 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 4.738272755 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.007977 \cdot 1.000000 \cdot 594}{1^2} \\ & \approx 4.738272755\end{aligned}$$
Modular invariants
Modular form 402930.2.a.ek
For more coefficients, see the Downloads section to the right.
| Modular degree: | 256608000 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $11$ | $I_{11}$ | split multiplicative | -1 | 1 | 11 | 11 |
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $11$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $37$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 \( 48840 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 36631 & 31086 \\ 3333 & 44419 \end{array}\right),\left(\begin{array}{rr} 48835 & 6 \\ 48834 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12022 & 32373 \\ 31625 & 42173 \end{array}\right),\left(\begin{array}{rr} 13319 & 0 \\ 0 & 48839 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 3961 & 31086 \\ 3003 & 44419 \end{array}\right),\left(\begin{array}{rr} 24421 & 31086 \\ 15543 & 44419 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 19537 & 31086 \\ 891 & 44419 \end{array}\right)$.
The torsion field $K:=\Q(E[48840])$ is a degree-$53200775282688000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48840\Z)$.
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 | $4$ | \( 201465 = 3^{2} \cdot 5 \cdot 11^{2} \cdot 37 \) |
| $3$ | additive | $8$ | \( 242 = 2 \cdot 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 80586 = 2 \cdot 3^{2} \cdot 11^{2} \cdot 37 \) |
| $11$ | additive | $62$ | \( 1665 = 3^{2} \cdot 5 \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 10890 = 2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 402930.ek
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 1110.n1, its twist by $33$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.