Properties

Label 1155.k1
Conductor $1155$
Discriminant $5.223\times 10^{14}$
j-invariant \( \frac{981281029968144361}{522287841796875} \)
CM no
Rank $1$
Torsion structure \(\Z/{4}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3+x^2-20702x-333459\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+x^2z-20702xz^2-333459z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-26830467x-15155409474\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, 1, 0, -20702, -333459])
 
gp: E = ellinit([1, 1, 0, -20702, -333459])
 
magma: E := EllipticCurve([1, 1, 0, -20702, -333459]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

\(\Z \oplus \Z/{4}\Z\)

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(212, 2099\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $1.1180524970622027663771016754$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(-68, 909\right) \) Copy content Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-68, 909\right) \), \( \left(-68, -841\right) \), \( \left(212, 2099\right) \), \( \left(212, -2311\right) \), \( \left(932, 27659\right) \), \( \left(932, -28591\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1155 \)  =  $3 \cdot 5 \cdot 7 \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $522287841796875 $  =  $3^{4} \cdot 5^{12} \cdot 7^{4} \cdot 11 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{981281029968144361}{522287841796875} \)  =  $3^{-4} \cdot 5^{-12} \cdot 7^{-4} \cdot 11^{-1} \cdot 47^{3} \cdot 21143^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.5154768544539158016888728531\dots$
Stable Faltings height: $1.5154768544539158016888728531\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.1180524970622027663771016754\dots$
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: $0.42289855051080424227399030465\dots$
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: $ 96 $  = $ 2\cdot( 2^{2} \cdot 3 )\cdot2^{2}\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $4$
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) $ ≈ $ 2.8369366824155446097475978416 $

Modular invariants

Modular form   1155.2.a.k

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);
 

\( q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + q^{7} - 3 q^{8} + q^{9} + q^{10} - q^{11} + q^{12} - 2 q^{13} + q^{14} - q^{15} - q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 4608
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is semistable. There are 4 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$(j)_{-}$
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$5$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$11$ $1$ $I_{1}$ Non-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
$2$ 2B 4.12.0.7

The image of the adelic Galois representation has level $3080$, index $48$, and genus $0$.

$p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ord nonsplit split split nonsplit ord ord ord ord ord ord ord ord ord ss
$\lambda$-invariant(s) 3 1 2 2 1 1 1 1 1 1 1 1 1 1 3,1
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,0

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 1155.k consists of 4 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{11}) \) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ 4.0.215600.2 \(\Z/8\Z\) Not in database
$8$ 8.0.116101021696.2 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.0.89991784960000.14 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.2.6227255754027.5 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/12\Z\) 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.