Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-76356208x+287870661588\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-76356208xz^2+287870661588z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-6184852875x+209876266856250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 115600 \) | = | $2^{4} \cdot 5^{2} \cdot 17^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-7314611834454016000000000$ | = | $-1 \cdot 2^{29} \cdot 5^{9} \cdot 17^{8} $ |
|
| j-invariant: | $j$ | = | \( -\frac{882216989}{131072} \) | = | $-1 \cdot 2^{-17} \cdot 17 \cdot 373^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5008568630717102209663856091$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.28817764785128775623443909086$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9569005082223869$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.686695412666291$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.071881542873366690465151474794$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot2\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.86257851448040028558181769752 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 0.862578514 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.071882 \cdot 1.000000 \cdot 12}{1^2} \\ & \approx 0.862578514\end{aligned}$$
Modular invariants
Modular form 115600.2.a.r
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24969600 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{21}^{*}$ | additive | -1 | 4 | 29 | 17 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $17$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|---|
| $17$ | 17B.4.2 | 17.72.1.2 | $72$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 680 = 2^{3} \cdot 5 \cdot 17 \), index $576$, genus $17$, and generators
$\left(\begin{array}{rr} 69 & 306 \\ 85 & 579 \end{array}\right),\left(\begin{array}{rr} 137 & 0 \\ 0 & 273 \end{array}\right),\left(\begin{array}{rr} 559 & 544 \\ 544 & 143 \end{array}\right),\left(\begin{array}{rr} 1 & 136 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 545 & 136 \\ 544 & 545 \end{array}\right),\left(\begin{array}{rr} 1 & 22 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 170 & 1 \end{array}\right),\left(\begin{array}{rr} 256 & 425 \\ 119 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 170 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 577 & 646 \\ 561 & 101 \end{array}\right),\left(\begin{array}{rr} 599 & 238 \\ 663 & 149 \end{array}\right),\left(\begin{array}{rr} 511 & 510 \\ 170 & 171 \end{array}\right)$.
The torsion field $K:=\Q(E[680])$ is a degree-$100270080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/680\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 1445 = 5 \cdot 17^{2} \) |
| $5$ | additive | $14$ | \( 4624 = 2^{4} \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 400 = 2^{4} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
17.
Its isogeny class 115600dh
consists of 2 curves linked by isogenies of
degree 17.
Twists
The minimal quadratic twist of this elliptic curve is 14450n1, its twist by $-340$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.11560.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.5345344000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $8$ | 8.8.1641354692000000.2 | \(\Z/17\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | add | ord | add | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 4 | - | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | - | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.