Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-1512778x-5760849116\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-1512778xz^2-5760849116z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-1960560315x-268772294675178\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 35378 \) | = | $2 \cdot 7^{2} \cdot 19^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-14114754528664572919808$ | = | $-1 \cdot 2^{27} \cdot 7^{6} \cdot 19^{7} $ |
|
| j-invariant: | $j$ | = | \( -\frac{69173457625}{2550136832} \) | = | $-1 \cdot 2^{-27} \cdot 5^{3} \cdot 19^{-1} \cdot 821^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9303441750681204635091520077$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.48516961095724358095196192003$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.054621266416818$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.582649042921267$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.054646201842886183471742823765$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 54 $ = $ 3^{3}\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.9508948995158539074741124833 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.950894900 \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 0.054646 \cdot 1.000000 \cdot 54}{1^2} \\ & \approx 2.950894900\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2449440 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $27$ | $I_{27}$ | split multiplicative | -1 | 1 | 27 | 27 |
| $7$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 |
|---|---|---|---|
| $3$ | 3B | 27.36.0.1 | $36$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 28728 = 2^{3} \cdot 3^{3} \cdot 7 \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 24623 & 0 \\ 0 & 28727 \end{array}\right),\left(\begin{array}{rr} 16444 & 16443 \\ 26145 & 3592 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 22906 & 21967 \end{array}\right),\left(\begin{array}{rr} 28675 & 54 \\ 28674 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5041 & 4179 \\ 13965 & 14456 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 15728 & 4095 \\ 27937 & 23960 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 7183 & 4158 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[28728])$ is a degree-$92643856220160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/28728\Z)$.
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 | $4$ | \( 17689 = 7^{2} \cdot 19^{2} \) |
| $3$ | good | $2$ | \( 17689 = 7^{2} \cdot 19^{2} \) |
| $7$ | additive | $26$ | \( 722 = 2 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 98 = 2 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 35378o
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 38a2, its twist by $133$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-399}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.152.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.3511808.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.7643717613.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.16716810419631.3 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.4065356736.6 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.774104572315466953767801.5 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.117072212150685561010672491665031168.2 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.1224616689352719226763839915372299581283631104.1 | \(\Z/18\Z\) | 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 | split | ord | ss | add | ord | ord | ord | add | ord | ord | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | 2 | 4 | 0,0 | - | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 |
| $\mu$-invariant(s) | 0 | 2 | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.