Properties

 Label 1805.b1 Conductor $1805$ Discriminant $705078125$ j-invariant $$\frac{7575076864}{1953125}$$ CM no Rank $0$ Torsion structure trivial

Related objects

Show commands: Magma / Pari/GP / SageMath

Simplified equation

 $$y^2+y=x^3-x^2-291x+1522$$ y^2+y=x^3-x^2-291x+1522 (homogenize, simplify) $$y^2z+yz^2=x^3-x^2z-291xz^2+1522z^3$$ y^2z+yz^2=x^3-x^2z-291xz^2+1522z^3 (dehomogenize, simplify) $$y^2=x^3-377568x+66493008$$ y^2=x^3-377568x+66493008 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 1, -291, 1522])

gp: E = ellinit([0, -1, 1, -291, 1522])

magma: E := EllipticCurve([0, -1, 1, -291, 1522]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

trivial

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

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(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $1.5047754359126224127052621482$

Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q + 2 q^{3} - 2 q^{4} - q^{5} - 4 q^{7} + q^{9} + 3 q^{11} - 4 q^{12} - 2 q^{13} - 2 q^{15} + 4 q^{16} + 6 q^{17} + O(q^{20})$$

 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:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

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

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

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
sage: gens = [[1, 18, 12, 217], [1, 18, 0, 1], [10, 9, 81, 73], [1, 18, 0, 191], [1625, 18, 558, 389], [1693, 18, 1692, 19], [1, 0, 18, 1], [10, 9, 1017, 1702], [13, 12, 1646, 1651]]

sage: GL(2,Integers(1710)).subgroup(gens)

magma: Gens := [[1, 18, 12, 217], [1, 18, 0, 1], [10, 9, 81, 73], [1, 18, 0, 191], [1625, 18, 558, 389], [1693, 18, 1692, 19], [1, 0, 18, 1], [10, 9, 1017, 1702], [13, 12, 1646, 1651]];

magma: sub<GL(2,Integers(1710))|Gens>;

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

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ss ord nonsplit ord ord ord ord add ss ord ord ord ord ord ord 0,9 0 0 0 0 0 0 - 0,0 0 0 0 0 0 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.

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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{57})$$ $$\Z/3\Z$$ 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.