Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1877949263x+31322978082906\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1877949263xz^2+31322978082906z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2433822244875x+1461441372769743750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 442225 \) | = | $5^{2} \cdot 7^{2} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-4502437830078125$ | = | $-1 \cdot 5^{9} \cdot 7^{2} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( -162677523113838677 \) | = | $-1 \cdot 7 \cdot 137^{3} \cdot 2083^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5474231743076388803055276514$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.54380689222295781849955231163$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0734398000116747$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.82126315539278$ | |||
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: | $\Omega$ | ≈ | $0.14465344503124021382866032075$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.57861378012496085531464128301 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.578613780 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.144653 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 0.578613780\end{aligned}$$
Modular invariants
Modular form 442225.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 58181760 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $19$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $37$ | 37B.8.2 | 37.114.4.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 98420 = 2^{2} \cdot 5 \cdot 7 \cdot 19 \cdot 37 \), index $2736$, genus $97$, and generators
$\left(\begin{array}{rr} 56981 & 41440 \\ 56980 & 56981 \end{array}\right),\left(\begin{array}{rr} 1 & 20748 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 41440 & 1 \end{array}\right),\left(\begin{array}{rr} 62245 & 39368 \\ 67488 & 29109 \end{array}\right),\left(\begin{array}{rr} 82993 & 19684 \\ 67488 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 49210 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 49210 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8436 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 78737 & 0 \\ 0 & 59053 \end{array}\right),\left(\begin{array}{rr} 56979 & 0 \\ 0 & 98419 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 67488 & 1 \end{array}\right),\left(\begin{array}{rr} 20388 & 28823 \\ 97717 & 32339 \end{array}\right),\left(\begin{array}{rr} 49211 & 49210 \\ 49210 & 49211 \end{array}\right),\left(\begin{array}{rr} 56241 & 0 \\ 0 & 14061 \end{array}\right),\left(\begin{array}{rr} 89985 & 8436 \\ 89984 & 89985 \end{array}\right),\left(\begin{array}{rr} 77595 & 43586 \\ 56943 & 25117 \end{array}\right),\left(\begin{array}{rr} 82955 & 82954 \\ 11951 & 12655 \end{array}\right)$.
The torsion field $K:=\Q(E[98420])$ is a degree-$7617383733657600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/98420\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 88445 = 5 \cdot 7^{2} \cdot 19^{2} \) |
| $5$ | additive | $14$ | \( 17689 = 7^{2} \cdot 19^{2} \) |
| $7$ | additive | $14$ | \( 9025 = 5^{2} \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
37.
Its isogeny class 442225v
consists of 2 curves linked by isogenies of
degree 37.
Twists
The minimal quadratic twist of this elliptic curve is 1225h2, its twist by $-95$.
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 $0$.