Properties

Label 105a2
Conductor $105$
Discriminant $11025$
j-invariant \( \frac{47045881}{11025} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -8, -7])
 
gp: E = ellinit([1, 0, 1, -8, -7])
 
magma: E := EllipticCurve([1, 0, 1, -8, -7]);
 

\(y^2+xy+y=x^3-8x-7\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z \times \Z/{2}\Z$

Torsion generators

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

\( \left(-1, 0\right) \), \( \left(3, -2\right) \)  Toggle raw display

Integral points

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

\( \left(-1, 0\right) \), \( \left(3, -2\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 105 \)  =  $3 \cdot 5 \cdot 7$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $11025 $  =  $3^{2} \cdot 5^{2} \cdot 7^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{47045881}{11025} \)  =  $3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 19^{6}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.51253574970719593248366541755\dots$
Stable Faltings height: $-0.51253574970719593248366541755\dots$

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: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $2.9317267268386014218882501760\dots$
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: $ 8 $  = $ 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) $ ≈ $ 1.4658633634193007109441250880108268142 $

Modular invariants

Modular form   105.2.a.a

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 8
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is semistable. There are 3 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$(j)_{-}$
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $2$ $I_{2}$ 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 4.12.0.1

$p$-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

$p$ 2 3 5 7
Reduction type ordinary split split split
$\lambda$-invariant(s) 0 3 1 1
$\mu$-invariant(s) 1 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 105a consists of 2 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{21}) \) \(\Z/2\Z \times \Z/4\Z\) 2.2.21.1-525.1-k3
$4$ \(\Q(i, \sqrt{5})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-5}, \sqrt{-21})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.31116960000.8 \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ 8.4.548903174400.13 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.2.265831216875.2 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) 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.