Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-7549x+388037\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-7549xz^2+388037z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-120787x+24713582\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{64541}{900}, \frac{21114263}{27000}\right) \) | $7.8116260334599188118305545223$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-1936230:21114263:27000]\) | $7.8116260334599188118305545223$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{64766}{225}, \frac{20146148}{3375}\right) \) | $7.8116260334599188118305545223$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 73034 \) | = | $2 \cdot 13 \cdot 53^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-36881496918656$ | = | $-1 \cdot 2^{7} \cdot 13 \cdot 53^{6} $ |
|
| j-invariant: | $j$ | = | \( -\frac{2146689}{1664} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 13^{-1} \cdot 43^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3019854083517888297155652678$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.68316054842427208735666930171$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.96783604338842$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.504976046208809$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.8116260334599188118305545223$ |
|
| Real period: | $\Omega$ | ≈ | $0.59707305424657146459386426141$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $9.3282228288598879670029824213 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 9.328222829 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.597073 \cdot 7.811626 \cdot 2}{1^2} \\ & \approx 9.328222829\end{aligned}$$
Modular invariants
Modular form 73034.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 288288 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $53$ | $2$ | $I_0^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $7$ | 7B.6.1 | 7.24.0.1 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 38584 = 2^{3} \cdot 7 \cdot 13 \cdot 53 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 28939 & 20034 \\ 0 & 6891 \end{array}\right),\left(\begin{array}{rr} 29681 & 742 \\ 34503 & 5195 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 38571 & 14 \\ 38570 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 13831 & 0 \\ 0 & 38583 \end{array}\right),\left(\begin{array}{rr} 9647 & 742 \\ 10017 & 5195 \end{array}\right),\left(\begin{array}{rr} 19293 & 742 \\ 371 & 5195 \end{array}\right)$.
The torsion field $K:=\Q(E[38584])$ is a degree-$6542153158754304$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/38584\Z)$.
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 | $4$ | \( 36517 = 13 \cdot 53^{2} \) |
| $7$ | good | $2$ | \( 36517 = 13 \cdot 53^{2} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 5618 = 2 \cdot 53^{2} \) |
| $53$ | additive | $1406$ | \( 26 = 2 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 73034d
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 26b1, its twist by $53$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{53}) \) | \(\Z/7\Z\) | 2.2.53.1-676.1-a2 |
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1610253632.1 | \(\Z/14\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $16$ | deg 16 | \(\Z/21\Z\) | 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 | 53 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | ss | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord | ss | ord | ord | add |
| $\lambda$-invariant(s) | 3 | 1,1 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | - |
| $\mu$-invariant(s) | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 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.