Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+2492x+23911\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+2492xz^2+23911z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+39877x+1570198\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-7, 81\right) \) | $2.9844496583254969943544194259$ | $\infty$ |
| \( \left(-\frac{37}{4}, \frac{33}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-7:81:1]\) | $2.9844496583254969943544194259$ | $\infty$ |
| \([-74:33:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-29, 624\right) \) | $2.9844496583254969943544194259$ | $\infty$ |
| \( \left(-38, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-7, 81\right) \), \( \left(-7, -75\right) \)
\([-7:81:1]\), \([-7:-75:1]\)
\((-29,\pm 624)\)
Invariants
| Conductor: | $N$ | = | \( 442090 \) | = | $2 \cdot 5 \cdot 11 \cdot 4019$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1250838835840$ | = | $-1 \cdot 2^{7} \cdot 5 \cdot 11^{2} \cdot 4019^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{1712108167716591}{1250838835840} \) | = | $2^{-7} \cdot 3^{3} \cdot 5^{-1} \cdot 11^{-2} \cdot 4019^{-2} \cdot 39877^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0095494046751420226998390008$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0095494046751420226998390008$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8366318472181958$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.6983442299457323$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.9844496583254969943544194259$ |
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| Real period: | $\Omega$ | ≈ | $0.54893831041038418582478712913$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 28 $ = $ 7\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.467951270622326852105774286 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.467951271 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.548938 \cdot 2.984450 \cdot 28}{2^2} \\ & \approx 11.467951271\end{aligned}$$
Modular invariants
Modular form 442090.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 543872 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $4019$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1768360 = 2^{3} \cdot 5 \cdot 11 \cdot 4019 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 2 & 1 \\ 884179 & 0 \end{array}\right),\left(\begin{array}{rr} 1048961 & 4 \\ 329562 & 9 \end{array}\right),\left(\begin{array}{rr} 353674 & 1 \\ 707343 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1768357 & 4 \\ 1768356 & 5 \end{array}\right),\left(\begin{array}{rr} 321521 & 4 \\ 643042 & 9 \end{array}\right),\left(\begin{array}{rr} 1547316 & 221049 \\ 1105225 & 663136 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[1768360])$ is a degree-$211538325547621416960000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1768360\Z)$.
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 | $4$ | \( 5 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 88418 = 2 \cdot 11 \cdot 4019 \) |
| $7$ | good | $2$ | \( 221045 = 5 \cdot 11 \cdot 4019 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40190 = 2 \cdot 5 \cdot 4019 \) |
| $4019$ | split multiplicative | $4020$ | \( 110 = 2 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 442090e
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
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.