Minimal Weierstrass equation
\( y^2 + x y + y = x^{3} - x^{2} + 475870 x - 171750378 \)
Mordell-Weil group structure
Torsion generators
\( \left(\frac{1211}{4}, -\frac{1215}{8}\right) \)
Integral points
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
|
|||||
| Conductor: | \( 190575 \) | = | \(3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}\) | ||
|
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
|
|||||
| Discriminant: | \(-19622342349928828125 \) | = | \(-1 \cdot 3^{10} \cdot 5^{7} \cdot 7^{4} \cdot 11^{6} \) | ||
|
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
|
|||||
| j-invariant: | \( \frac{590589719}{972405} \) | = | \(3^{-4} \cdot 5^{-1} \cdot 7^{-4} \cdot 839^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
|
magma: Rank(E);
sage: E.rank()
|
|||
| Rank: | \(0\) | ||
|
magma: Regulator(E);
sage: E.regulator()
|
|||
| Regulator: | \(1\) | ||
|
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
|
|||
| Real period: | \(0.114117308071\) | ||
|
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
|
|||
| Tamagawa product: | \( 64 \) = \( 2^{2}\cdot2\cdot2^{2}\cdot2 \) | ||
|
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
|
|||
| Torsion order: | \(2\) | ||
|
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
|
|||
| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 190575.2.a.bp
|
magma: ModularDegree(E);
sage: E.modular_degree()
|
|||
| Modular degree: | 4423680 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 1.82587692913 \)
Local data
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(4\) | \( I_4^{*} \) | Additive | -1 | 2 | 10 | 4 |
| \(5\) | \(2\) | \( I_1^{*} \) | Additive | 1 | 2 | 7 | 1 |
| \(7\) | \(4\) | \( I_{4} \) | Split multiplicative | -1 | 1 | 4 | 4 |
| \(11\) | \(2\) | \( I_0^{*} \) | Additive | -1 | 2 | 6 | 0 |
Galois representations
The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | B |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class 190575be
consists of 4 curves linked by isogenies of
degrees dividing 4.
Growth of torsion in number fields
The number fields $K$ of degree up to 7 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 | \(\Q(\sqrt{-33}) \) | \(\Z/4\Z\) | Not in database |
| \(\Q(\sqrt{165}) \) | \(\Z/4\Z\) | Not in database | |
| \(\Q(\sqrt{-5}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database | |
| 4 | \(\Q(\sqrt{-5}, \sqrt{-33})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.