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Properties

Label 1155.e2
Conductor $1155$
Discriminant $14707625625$
j-invariant \( \frac{74093292126001}{14707625625} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z \oplus \Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3+x^2-875x-8440\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3+x^2z-875xz^2-8440z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1134027x-376757946\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, 1, 1, -875, -8440])
 
gp: E = ellinit([1, 1, 1, -875, -8440])
 
magma: E := EllipticCurve([1, 1, 1, -875, -8440]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(-22, 38\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.85009030288104411292552745618$

Torsion generators

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

\( \left(-11, 5\right) \), \( \left(33, -17\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-22, 38\right) \), \( \left(-22, -17\right) \), \( \left(-12, 28\right) \), \( \left(-12, -17\right) \), \( \left(-11, 5\right) \), \( \left(33, -17\right) \), \( \left(38, 103\right) \), \( \left(38, -142\right) \), \( \left(133, 1433\right) \), \( \left(133, -1567\right) \), \( \left(528, 11863\right) \), \( \left(528, -12392\right) \) 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: $14707625625 $  =  $3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{74093292126001}{14707625625} \)  =  $3^{-4} \cdot 5^{-4} \cdot 7^{-4} \cdot 11^{-2} \cdot 97^{3} \cdot 433^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.66729310237448382314064497860\dots$
Stable Faltings height: $0.66729310237448382314064497860\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.85009030288104411292552745618\dots$
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: $0.88997304932837026230710109381\dots$
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: $ 32 $  = $ 2\cdot2^{2}\cdot2\cdot2 $
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) $ ≈ $ 1.5131149181190413784512069043 $

Modular invariants

Modular form   1155.2.a.e

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: 1024
$ \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
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$7$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$11$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

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
$2$ 2Cs 4.24.0.6
sage: gens = [[9233, 8, 9232, 9], [8407, 2, 822, 9235], [1, 0, 8, 1], [6161, 8, 6164, 33], [1, 8, 0, 1], [5281, 4, 0, 1], [5, 4, 9236, 9237], [4625, 6932, 6952, 4629], [7, 6936, 9234, 2305], [3697, 8, 5548, 33]]
 
sage: GL(2,Integers(9240)).subgroup(gens)
 
magma: Gens := [[9233, 8, 9232, 9], [8407, 2, 822, 9235], [1, 0, 8, 1], [6161, 8, 6164, 33], [1, 8, 0, 1], [5281, 4, 0, 1], [5, 4, 9236, 9237], [4625, 6932, 6952, 4629], [7, 6936, 9234, 2305], [3697, 8, 5548, 33]];
 
magma: sub<GL(2,Integers(9240))|Gens>;
 

The image of the adelic Galois representation has level $9240$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 9233 & 8 \\ 9232 & 9 \end{array}\right),\left(\begin{array}{rr} 8407 & 2 \\ 822 & 9235 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 6161 & 8 \\ 6164 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5281 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 9236 & 9237 \end{array}\right),\left(\begin{array}{rr} 4625 & 6932 \\ 6952 & 4629 \end{array}\right),\left(\begin{array}{rr} 7 & 6936 \\ 9234 & 2305 \end{array}\right),\left(\begin{array}{rr} 3697 & 8 \\ 5548 & 33 \end{array}\right)$

$p$-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 nonsplit split ord ord ord ord ord ord ord ord ord ord
$\lambda$-invariant(s) 1 1 2 3 2 1 1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) 2 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.e consists of 6 curves linked by isogenies of degrees dividing 8.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ 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
$2$ \(\Q(\sqrt{11}) \) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-1}) \) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(i, \sqrt{11})\) \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.1088900598016.3 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.4.5877614223360000.10 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.0.7965941760000.3 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ deg 8 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.2.3892034846266875.7 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$16$ deg 16 \(\Z/4\Z \oplus \Z/8\Z\) Not in database
$16$ deg 16 \(\Z/4\Z \oplus \Z/8\Z\) Not in database
$16$ deg 16 \(\Z/4\Z \oplus \Z/8\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/12\Z\) 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.