Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2+6135403x-6798525555\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z+6135403xz^2-6798525555z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+98166453x-435007469050\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1145, 40980)$ | $3.6463488956943716258428869888$ | $\infty$ |
| $(3851/4, -3855/8)$ | $0$ | $2$ |
Integral points
\( \left(1145, 40980\right) \), \( \left(1145, -42126\right) \), \( \left(49877, 11127510\right) \), \( \left(49877, -11177388\right) \)
Invariants
| Conductor: | $N$ | = | \( 106722 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-34739196977520118990368$ | = | $-1 \cdot 2^{5} \cdot 3^{16} \cdot 7^{6} \cdot 11^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{168105213359}{228637728} \) | = | $2^{-5} \cdot 3^{-10} \cdot 11^{-2} \cdot 5519^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0109167557203668958047828274$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.28970790045947012552351204823$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1002106061618209$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.0788327699663185$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.6463488956943716258428869888$ |
|
| Real period: | $\Omega$ | ≈ | $0.061865856003421548009098930047$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ 5\cdot2^{2}\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $9.0233798287705269127770033780 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 9.023379829 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.061866 \cdot 3.646349 \cdot 160}{2^2} \\ & \approx 9.023379829\end{aligned}$$
Modular invariants
Modular form 106722.2.a.eg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 13824000 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $3$ | $4$ | $I_{10}^{*}$ | additive | -1 | 2 | 16 | 10 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $2$ | 2B | 8.6.0.5 |
| $5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 3079 & 6594 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 9221 & 20 \\ 9220 & 21 \end{array}\right),\left(\begin{array}{rr} 5286 & 2653 \\ 7175 & 7736 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 666 & 5285 \\ 385 & 1366 \end{array}\right),\left(\begin{array}{rr} 5811 & 2660 \\ 6580 & 2507 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 9000 & 8891 \end{array}\right),\left(\begin{array}{rr} 3959 & 0 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 6719 & 6594 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$3269984256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $4$ | \( 53361 = 3^{2} \cdot 7^{2} \cdot 11^{2} \) |
| $3$ | additive | $8$ | \( 11858 = 2 \cdot 7^{2} \cdot 11^{2} \) |
| $5$ | good | $2$ | \( 53361 = 3^{2} \cdot 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $26$ | \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 106722hn
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 66c2, its twist by $-231$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-231}) \) | \(\Z/10\Z\) | not in database |
| $4$ | 4.2.1707552.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-231})\) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | 8.0.11942845778755584.481 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.186606965293056.58 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.2915733832704.30 | \(\Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/30\Z\) | not in database |
| $20$ | 20.4.2830162324148238464682513639699737548828125.1 | \(\Z/10\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | ord | add | add | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 3 | - | - | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | - | 0 | - | - | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.