Properties

 Label 405.b2 Conductor $405$ Discriminant $-295245$ j-invariant $$-\frac{9}{5}$$ CM no Rank $1$ Torsion structure trivial

Related objects

Show commands: Magma / Pari/GP / SageMath

Simplified equation

 $$y^2+xy+y=x^3-x^2-2x-26$$ y^2+xy+y=x^3-x^2-2x-26 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-2xz^2-26z^3$$ y^2z+xyz+yz^2=x^3-x^2z-2xz^2-26z^3 (dehomogenize, simplify) $$y^2=x^3-27x-1674$$ y^2=x^3-27x-1674 (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 1, -2, -26])

gp: E = ellinit([1, -1, 1, -2, -26])

magma: E := EllipticCurve([1, -1, 1, -2, -26]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

Mordell-Weil group structure

$$\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(4, 2\right)$$ (4, 2) $\hat{h}(P)$ ≈ $0.30602925338795851320903564775$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(4, 2\right)$$, $$\left(4, -7\right)$$, $$\left(13, 38\right)$$, $$\left(13, -52\right)$$, $$\left(14, 43\right)$$, $$\left(14, -58\right)$$, $$\left(9094, 862643\right)$$, $$\left(9094, -871738\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$405$$ = $3^{4} \cdot 5$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-295245$ = $-1 \cdot 3^{10} \cdot 5$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{9}{5}$$ = $-1 \cdot 3^{2} \cdot 5^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.27128371394435064486856185418\dots$ Stable Faltings height: $-1.1867939545011087210312662183\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.30602925338795851320903564775\dots$ 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: $1.3820485347634765158428338742\dots$ 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: $3$  = $3\cdot1$ 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: $L'(E,1)$ ≈ $1.2688418437187662336679690472$

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);

$$q - q^{2} - q^{4} + q^{5} - 3 q^{7} + 3 q^{8} - q^{10} + 2 q^{11} - 2 q^{13} + 3 q^{14} - q^{16} - 4 q^{17} - 8 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 36 $\Gamma_0(N)$-optimal: yes 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$(j)_{-}$
$3$ $3$ $IV^{*}$ Additive -1 4 10 0
$5$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

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
$7$ 7B 7.8.0.1

The image of the adelic Galois representation has level $1260$, index $96$, and genus $2$.

$p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

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 ord add split ord ord ord ord ord ord ord ss ord ord ord ord 1 - 2 1 1 1 3 1 1 1 1,1 1 1 1 1 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 $d=$ 7.
Its isogeny class 405.b consists of 2 curves linked by isogenies of degree 7.

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 $K$ $3$ 3.1.1620.1 $$\Z/2\Z$$ Not in database $6$ 6.0.52488000.2 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ $$\Q(\zeta_{9})$$ $$\Z/7\Z$$ Not in database $8$ 8.2.110716875.2 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $18$ 18.0.488038239039168000000.1 $$\Z/14\Z$$ 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.