Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-102704x+11705006\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-102704xz^2+11705006z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-133103763x+546508082862\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-367, 183)$ | $0$ | $2$ |
| $(225, -113)$ | $0$ | $2$ |
Integral points
\( \left(-367, 183\right) \), \( \left(225, -113\right) \)
Invariants
| Conductor: | $N$ | = | \( 451770 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $10058673813843600$ | = | $2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2} \cdot 37^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{46694890801}{3920400} \) | = | $2^{-4} \cdot 3^{-4} \cdot 5^{-2} \cdot 11^{-2} \cdot 13^{3} \cdot 277^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8124004162445705454530103642$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.0069414599224583232689625286837$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9379136566213516$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5506229942600274$ | |||
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: | $\Omega$ | ≈ | $0.39765961300653078754690959870$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2^{2}\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.1812769040522463003752767896 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.181276904 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.397660 \cdot 1.000000 \cdot 128}{4^2} \\ & \approx 3.181276904\end{aligned}$$
Modular invariants
Modular form 451770.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3317760 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $37$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 |
|---|---|---|
| $2$ | 2Cs | 4.12.0.3 |
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 $192$, genus $1$, and generators
$\left(\begin{array}{rr} 40923 & 20794 \\ 15836 & 35965 \end{array}\right),\left(\begin{array}{rr} 48833 & 8 \\ 48832 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 29527 & 25086 \\ 5994 & 23755 \end{array}\right),\left(\begin{array}{rr} 8585 & 4292 \\ 4292 & 24421 \end{array}\right),\left(\begin{array}{rr} 32561 & 17168 \\ 16724 & 19833 \end{array}\right),\left(\begin{array}{rr} 15319 & 40922 \\ 12654 & 23755 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 48836 & 48837 \end{array}\right),\left(\begin{array}{rr} 44879 & 0 \\ 0 & 48839 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[48840])$ is a degree-$4433397940224000$ 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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 1369 = 37^{2} \) |
| $3$ | split multiplicative | $4$ | \( 150590 = 2 \cdot 5 \cdot 11 \cdot 37^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 90354 = 2 \cdot 3 \cdot 11 \cdot 37^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 41070 = 2 \cdot 3 \cdot 5 \cdot 37^{2} \) |
| $37$ | additive | $686$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 451770t
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 330b2, its twist by $37$.
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 $0$.