Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-17204x-864798\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-17204xz^2-864798z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-22295763x-40281116562\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(20309/121, 1207345/1331)$ | $9.5979851103609487284311730946$ | $\infty$ |
| $(-71, 35)$ | $0$ | $2$ |
Integral points
\( \left(-71, 35\right) \)
Invariants
| Conductor: | $N$ | = | \( 68770 \) | = | $2 \cdot 5 \cdot 13 \cdot 23^{2}$ |
|
| Discriminant: | $\Delta$ | = | $3848933114000$ | = | $2^{4} \cdot 5^{3} \cdot 13 \cdot 23^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{3803721481}{26000} \) | = | $2^{-4} \cdot 5^{-3} \cdot 7^{3} \cdot 13^{-1} \cdot 223^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2488903661081835852955902421$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.31885674185639126010778617380$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.906187683689355$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6694462532387893$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.5979851103609487284311730946$ |
|
| Real period: | $\Omega$ | ≈ | $0.41703654201410039677729537414$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.0027105207277538278628960197 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.002710521 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.417037 \cdot 9.597985 \cdot 4}{2^2} \\ & \approx 4.002710521\end{aligned}$$
Modular invariants
Modular form 68770.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 285120 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $23$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.3 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 35880 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 23 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 35857 & 24 \\ 35856 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 26911 & 9384 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 492 & 1687 \end{array}\right),\left(\begin{array}{rr} 11961 & 1564 \\ 19780 & 31281 \end{array}\right),\left(\begin{array}{rr} 1266 & 9361 \\ 32591 & 29648 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1559 & 0 \\ 0 & 35879 \end{array}\right),\left(\begin{array}{rr} 7936 & 28083 \\ 6141 & 10834 \end{array}\right),\left(\begin{array}{rr} 17941 & 9384 \\ 17940 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 35780 & 35861 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[35880])$ is a degree-$645298693079040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/35880\Z)$.
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 | $4$ | \( 34385 = 5 \cdot 13 \cdot 23^{2} \) |
| $3$ | good | $2$ | \( 13754 = 2 \cdot 13 \cdot 23^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 13754 = 2 \cdot 13 \cdot 23^{2} \) |
| $13$ | split multiplicative | $14$ | \( 5290 = 2 \cdot 5 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 130 = 2 \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 68770.a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 130.a2, its twist by $-23$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.550160.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-23}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.150120728784.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.1278806208160000.25 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.302676025600.3 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.35576231699885585492398632402937500000000.3 | \(\Z/18\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | ord | nonsplit | ord | ord | split | ord | ord | add | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 3 | 1 | 1 | 1 | 2 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.