Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-42067x+2662574\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-42067xz^2+2662574z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-54518211x+124388618814\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(43519/225, 4163792/3375)$ | $7.7689318155892902342442717128$ | $\infty$ |
| $(159, -80)$ | $0$ | $2$ |
Integral points
\( \left(159, -80\right) \)
Invariants
| Conductor: | $N$ | = | \( 422142 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $1693549518753792$ | = | $2^{12} \cdot 3 \cdot 7^{2} \cdot 19 \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{55611739513}{11440128} \) | = | $2^{-12} \cdot 3^{-1} \cdot 7^{-2} \cdot 11^{3} \cdot 19^{-1} \cdot 347^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6374747011249084377892672796$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.069727593160333592385890863695$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9136325031445056$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.362487392417616$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.7689318155892902342442717128$ |
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| Real period: | $\Omega$ | ≈ | $0.44731570618736774721280000890$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.9503304428236648394133906515 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.950330443 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.447316 \cdot 7.768932 \cdot 8}{2^2} \\ & \approx 6.950330443\end{aligned}$$
Modular invariants
Modular form 422142.2.a.bc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2162688 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $23$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10488 = 2^{3} \cdot 3 \cdot 19 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 4739 & 4278 \\ 7130 & 1427 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 455 & 0 \\ 0 & 10487 \end{array}\right),\left(\begin{array}{rr} 10481 & 8 \\ 10480 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 10482 & 10483 \end{array}\right),\left(\begin{array}{rr} 5176 & 8211 \\ 10189 & 2738 \end{array}\right),\left(\begin{array}{rr} 10420 & 2737 \\ 2231 & 3198 \end{array}\right),\left(\begin{array}{rr} 1657 & 1656 \\ 2806 & 7591 \end{array}\right)$.
The torsion field $K:=\Q(E[10488])$ is a degree-$50524760309760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10488\Z)$.
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 | $4$ | \( 30153 = 3 \cdot 19 \cdot 23^{2} \) |
| $3$ | split multiplicative | $4$ | \( 70357 = 7 \cdot 19 \cdot 23^{2} \) |
| $7$ | split multiplicative | $8$ | \( 60306 = 2 \cdot 3 \cdot 19 \cdot 23^{2} \) |
| $19$ | split multiplicative | $20$ | \( 22218 = 2 \cdot 3 \cdot 7 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 422142.bc
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 798.e3, its twist by $-23$.
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.