Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-22503407x-75735722176\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-22503407xz^2-75735722176z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-29164414851x-3533438360587266\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(23751/4, -23755/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 422142 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1748454760139583323692998$ | = | $-1 \cdot 2 \cdot 3^{4} \cdot 7^{2} \cdot 19 \cdot 23^{14} $ |
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| j-invariant: | $j$ | = | \( -\frac{8513369695913641273}{11811019422050982} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-4} \cdot 7^{-2} \cdot 11^{3} \cdot 13^{3} \cdot 19^{-1} \cdot 23^{-8} \cdot 109^{3} \cdot 131^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3441338465702198419732221548$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7763867386056449965698457389$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9584747671715312$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.911885352679754$ | |||
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: | $\Omega$ | ≈ | $0.032986346728160230208878485523$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2^{2}\cdot2\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.3750169644275365750392509576 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.375016964 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.032986 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 2.375016964\end{aligned}$$
Modular invariants
Modular form 422142.2.a.z
For more coefficients, see the Downloads section to the right.
| Modular degree: | 67043328 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $23$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
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
$\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 10482 & 10483 \end{array}\right),\left(\begin{array}{rr} 455 & 0 \\ 0 & 10487 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 10481 & 8 \\ 10480 & 9 \end{array}\right),\left(\begin{array}{rr} 4739 & 9292 \\ 9338 & 4509 \end{array}\right),\left(\begin{array}{rr} 3497 & 920 \\ 3956 & 3681 \end{array}\right),\left(\begin{array}{rr} 2600 & 8211 \\ 5957 & 2738 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 9408 & 10097 \\ 5635 & 9270 \end{array}\right)$.
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 | $4$ | \( 10051 = 19 \cdot 23^{2} \) |
| $3$ | split multiplicative | $4$ | \( 140714 = 2 \cdot 7 \cdot 19 \cdot 23^{2} \) |
| $7$ | nonsplit 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 | $288$ | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 422142.z
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 18354.n3, its twist by $-23$.
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 $0$.