↑

Properties

Label 15.a1
Conductor $15$
Discriminant $405$
j-invariant \( \frac{1114544804970241}{405} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -2160, -39540])
 
gp: E = ellinit([1, 1, 1, -2160, -39540])
 
magma: E := EllipticCurve([1, 1, 1, -2160, -39540]);
 

\(y^2+xy+y=x^3+x^2-2160x-39540\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

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

\( \left(-\frac{109}{4}, \frac{105}{8}\right) \)  Toggle raw display

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 15 \)  =  $3 \cdot 5$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $405 $  =  $3^{4} \cdot 5 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{1114544804970241}{405} \)  =  $3^{-4} \cdot 5^{-1} \cdot 103681^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.29086960847710152197363562429\dots$
Stable Faltings height: $0.29086960847710152197363562429\dots$

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: $0.70030152116630101159009041840\dots$
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: $ 2 $  = $ 2\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
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) $ ≈ $ 0.35015076058315050579504520920239987269 $

Modular invariants

Modular form   15.2.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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 4
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is 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$(j)_{-}$
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$5$ $1$ $I_{1}$ 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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 16.96.0.148

$p$-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
Reduction type ordinary nonsplit split
$\lambda$-invariant(s) 0 0 1
$\mu$-invariant(s) 3 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, 4, 8 and 16.
Its isogeny class 15.a consists of 5 curves linked by isogenies of degrees dividing 16.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{5}) \) \(\Z/2\Z \times \Z/2\Z\) 2.2.5.1-45.1-a9
$2$ \(\Q(\sqrt{-5}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-1}) \) \(\Z/4\Z\) 2.0.4.1-225.2-a10
$4$ \(\Q(i, \sqrt{5})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ 4.2.2000.1 \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ 4.0.32000.1 \(\Z/8\Z\) Not in database
$4$ \(\Q(\zeta_{8})\) \(\Z/8\Z\) Not in database
$4$ \(\Q(i, \sqrt{10})\) \(\Z/8\Z\) Not in database
$8$ 8.0.64000000.3 \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ 8.0.1024000000.6 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.0.40960000.1 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.2.414720000000.4 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.0.33973862400.10 \(\Z/16\Z\) Not in database
$8$ 8.0.21233664000000.7 \(\Z/16\Z\) Not in database
$8$ 8.0.21233664000000.8 \(\Z/16\Z\) Not in database
$8$ 8.2.110716875.2 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ 16.0.16777216000000000000.3 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ 16.0.450868486864896000000000000.3 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) 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.