Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-420468901x+216442550194448\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-420468901xz^2+216442550194448z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-544927695075x+10098345256655262750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1053907, 1081309046\right) \) | $5.1413825933186894865939274888$ | $\infty$ |
| \( \left(-\frac{249497}{4}, \frac{249493}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1053907:1081309046:1]\) | $5.1413825933186894865939274888$ | $\infty$ |
| \([-498994:249493:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37940655, 233676576000\right) \) | $5.1413825933186894865939274888$ | $\infty$ |
| \( \left(-2245470, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(1053907, 1081309046\right) \), \( \left(1053907, -1082362954\right) \)
\([1053907:1081309046:1]\), \([1053907:-1082362954:1]\)
\((37940655,\pm 233676576000)\)
Invariants
| Conductor: | $N$ | = | \( 126150 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 29^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-20233316116675980647787600000000$ | = | $-1 \cdot 2^{10} \cdot 3^{20} \cdot 5^{8} \cdot 29^{9} $ |
|
| j-invariant: | $j$ | = | \( -\frac{36267977929301}{89261680665600} \) | = | $-1 \cdot 2^{-10} \cdot 3^{-20} \cdot 5^{-2} \cdot 79^{3} \cdot 419^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.6860301152407799788431168804$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3558392865338742711552831895$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.2035853381343617$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.772098888731977$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.1413825933186894865939274888$ |
|
| Real period: | $\Omega$ | ≈ | $0.017372851704809967621891857825$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 320 $ = $ 2\cdot( 2^{2} \cdot 5 )\cdot2^{2}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $7.1456381881133509675653384036 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 7.145638188 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.017373 \cdot 5.141383 \cdot 320}{2^2} \\ & \approx 7.145638188\end{aligned}$$
Modular invariants
Modular form 126150.2.a.bk
For more coefficients, see the Downloads section to the right.
| Modular degree: | 311808000 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $29$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $5$ | 5B | 5.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 580 = 2^{2} \cdot 5 \cdot 29 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 6 & 5 \\ 345 & 46 \end{array}\right),\left(\begin{array}{rr} 561 & 20 \\ 560 & 21 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 340 & 231 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 518 & 565 \\ 215 & 262 \end{array}\right),\left(\begin{array}{rr} 463 & 574 \\ 0 & 579 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[580])$ is a degree-$109132800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/580\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 725 = 5^{2} \cdot 29 \) |
| $3$ | split multiplicative | $4$ | \( 42050 = 2 \cdot 5^{2} \cdot 29^{2} \) |
| $5$ | additive | $18$ | \( 841 = 29^{2} \) |
| $29$ | additive | $254$ | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 126150.bk
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 25230.h2, its twist by $145$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-29}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{58 -6 \sqrt{145}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{290 +2 \sqrt{145}})\) | \(\Z/10\Z\) | not in database |
| $8$ | 8.0.1522747701760000.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.95171731360000.28 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.2379293284000000.12 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | 8.4.148705830250000.8 | \(\Z/20\Z\) | 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 |
| $20$ | 20.0.3139280578615647551832444150932133197784423828125.2 | \(\Z/10\Z\) | 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 | nonsplit | split | add | ord | ss | ord | ord | ord | ord | add | ss | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 3 | 4 | - | 1 | 1,1 | 1 | 3 | 1 | 1 | - | 1,1 | 1 | 1,1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | - | 0 | 0,0 | 0 | 0 | 0 | 0 | - | 0,0 | 0 | 0,0 | 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.