Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-11502482x+10183601495\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-11502482xz^2+10183601495z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-14907216051x+475170833010510\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(855, 30784)$ | $3.1216849155603944784744654026$ | $\infty$ |
| $(-3769, 1884)$ | $0$ | $2$ |
| $(963, -482)$ | $0$ | $2$ |
Integral points
\( \left(-3769, 1884\right) \), \( \left(-3405, 101074\right) \), \( \left(-3405, -97670\right) \), \( \left(855, 30784\right) \), \( \left(855, -31640\right) \), \( \left(963, -482\right) \)
Invariants
| Conductor: | $N$ | = | \( 422331 \) | = | $3 \cdot 7^{2} \cdot 13^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $52589710709395698524049$ | = | $3^{8} \cdot 7^{6} \cdot 13^{8} \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{296380748763217}{92608836489} \) | = | $3^{-8} \cdot 13^{-2} \cdot 17^{-4} \cdot 61^{3} \cdot 1093^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0640121082413302397326984472$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.80858235498290521915327835470$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9639015803583432$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.661876324016547$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1216849155603944784744654026$ |
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| Real period: | $\Omega$ | ≈ | $0.10386698124982868720319637909$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2^{3}\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.1878398174781526714349840035 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.187839817 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.103867 \cdot 3.121685 \cdot 256}{4^2} \\ & \approx 5.187839817\end{aligned}$$
Modular invariants
Modular form 422331.2.a.bq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 33030144 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $17$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 8.24.0.13 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12376 = 2^{3} \cdot 7 \cdot 13 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 4425 & 2212 \\ 5768 & 2661 \end{array}\right),\left(\begin{array}{rr} 3641 & 10612 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 12372 & 12373 \end{array}\right),\left(\begin{array}{rr} 10737 & 7070 \\ 3962 & 3541 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 12369 & 8 \\ 12368 & 9 \end{array}\right),\left(\begin{array}{rr} 5303 & 0 \\ 0 & 12375 \end{array}\right),\left(\begin{array}{rr} 3543 & 448 \\ 8834 & 11929 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[12376])$ is a degree-$33111266033664$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12376\Z)$.
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 | $2$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 140777 = 7^{2} \cdot 13^{2} \cdot 17 \) |
| $7$ | additive | $26$ | \( 8619 = 3 \cdot 13^{2} \cdot 17 \) |
| $13$ | additive | $98$ | \( 2499 = 3 \cdot 7^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 24843 = 3 \cdot 7^{2} \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 422331.bq
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 663.a2, its twist by $-91$.
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.