Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+8277870x+1819757052\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+8277870xz^2+1819757052z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+10728119493x+84870400659606\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(954, 102408)$ | $0$ | $3$ |
Integral points
\( \left(954, 102408\right) \), \( \left(954, -103362\right) \)
Invariants
| Conductor: | $N$ | = | \( 92910 \) | = | $2 \cdot 3 \cdot 5 \cdot 19 \cdot 163$ |
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| Discriminant: | $\Delta$ | = | $-37731927035519394651000$ | = | $-1 \cdot 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 19^{9} \cdot 163^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{62730610483865150939136479}{37731927035519394651000} \) | = | $2^{-3} \cdot 3^{-3} \cdot 5^{-3} \cdot 7^{3} \cdot 13^{6} \cdot 19^{-9} \cdot 43^{3} \cdot 73^{3} \cdot 107^{3} \cdot 163^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0213343449279879654558488641$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.0213343449279879654558488641$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0472465811977227$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.192664086119918$ | |||
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: | $\Omega$ | ≈ | $0.070731178042710170971931543731$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 729 $ = $ 3\cdot3\cdot3\cdot3^{2}\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.7292254214595238487264550422 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 5.729225421 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.070731 \cdot 1.000000 \cdot 729}{3^2} \\ & \approx 5.729225421\end{aligned}$$
Modular invariants
Modular form 92910.2.a.bd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10182672 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $19$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $163$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $3$ | 3Cs.1.1 | 9.72.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1114920 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \cdot 163 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 891937 & 18 \\ 222993 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 836191 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 184681 & 18 \\ 547209 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 704161 & 18 \\ 762849 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 278731 & 557478 \\ 278739 & 588593 \end{array}\right),\left(\begin{array}{rr} 557461 & 18 \\ 557469 & 163 \end{array}\right),\left(\begin{array}{rr} 1114903 & 18 \\ 1114902 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[1114920])$ is a degree-$1719437149051394457600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1114920\Z)$.
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 | $4$ | \( 46455 = 3 \cdot 5 \cdot 19 \cdot 163 \) |
| $3$ | split multiplicative | $4$ | \( 1 \) |
| $5$ | split multiplicative | $6$ | \( 18582 = 2 \cdot 3 \cdot 19 \cdot 163 \) |
| $19$ | split multiplicative | $20$ | \( 4890 = 2 \cdot 3 \cdot 5 \cdot 163 \) |
| $163$ | split multiplicative | $164$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 92910.bd
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $3$ | 3.1.371640.1 | \(\Z/6\Z\) | not in database |
| $3$ | 3.1.24300.2 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.51329537866944000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.1771470000.4 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $6$ | 6.0.414348868800.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $9$ | 9.1.818358628006397589120000000.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.1955476758561888850288369070881021250289363000000000000.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.2682409819700807750738503526585763031947.4 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.2009132532097540285528628351599645027123200000000000000.1 | \(\Z/3\Z \oplus \Z/18\Z\) | not in database |
| $18$ | 18.0.38195497920744361334000836603708950465107020871368704000000000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 19 | 163 |
|---|---|---|---|---|---|
| Reduction type | split | split | split | split | split |
| $\lambda$-invariant(s) | 4 | 5 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 1 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.