Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+y=x^3-x^2-291x+1522\)
|
(homogenize, simplify) |
|
\(y^2z+yz^2=x^3-x^2z-291xz^2+1522z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-377568x+66493008\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 1805 \) | = | $5 \cdot 19^{2}$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $705078125 $ | = | $5^{9} \cdot 19^{2} $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{7575076864}{1953125} \) | = | $2^{15} \cdot 5^{-9} \cdot 19 \cdot 23^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.40708034707446269682927796809\dots$ | ||
| Stable Faltings height: | $-0.083659482786610713172226603891\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
| |||
| Real period: | $1.5047754359126224127052621482\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
| |||
| Tamagawa product: | $ 1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
| |||
| Torsion order: | $1$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
| |||
| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
| |||
| Special value: | $ L(E,1) $ ≈ $ 1.5047754359126224127052621482 $ | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 756 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $5$ | $1$ | $I_{9}$ | Non-split multiplicative | 1 | 1 | 9 | 9 |
| $19$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 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 |
|---|---|---|
| $3$ | 3B | 9.12.0.2 |
The image of the adelic Galois representation has level $1710$, index $144$, genus $2$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 12 & 217 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 191 \end{array}\right),\left(\begin{array}{rr} 1625 & 18 \\ 558 & 389 \end{array}\right),\left(\begin{array}{rr} 1693 & 18 \\ 1692 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 1017 & 1702 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 1646 & 1651 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ord | nonsplit | ord | ord | ord | ord | add | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 0,9 | 0 | 0 | 0 | 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 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 1805.b
consists of 2 curves linked by isogenies of
degree 3.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/3\Z\) | Not in database |
| $3$ | 3.3.7220.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.6.260642000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $6$ | 6.0.22284891.1 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.6.26741869200.1 | \(\Z/6\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | 12.0.4469547301936929.2 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $18$ | 18.6.115765164098281427122348305637113.3 | \(\Z/9\Z\) | Not in database |
| $18$ | 18.0.708290662705838589438144000000.1 | \(\Z/6\Z\) | 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.