Properties

Label 225.b1
Conductor $225$
Discriminant $4613203125$
j-invariant \( \frac{1114544804970241}{405} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -486005, 130530872])
 
gp: E = ellinit([1, -1, 1, -486005, 130530872])
 
magma: E := EllipticCurve([1, -1, 1, -486005, 130530872]);
 

\(y^2+xy+y=x^3-x^2-486005x+130530872\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

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

\( \left(\frac{1611}{4}, -\frac{1615}{8}\right) \)  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: \( 225 \)  =  $3^{2} \cdot 5^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $4613203125 $  =  $3^{10} \cdot 5^{7} $
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: $1.6448947090282065549716379094\dots$
Stable Faltings height: $0.29086960847710152197363562433\dots$

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: $0.82429593904393569890541103843\dots$
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: $ 4 $  = $ 2\cdot2 $
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) $ ≈ $ 0.82429593904393569890541103843228277079 $

Modular invariants

Modular form   225.2.a.b

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);
 

\( q - q^{2} - q^{4} + 3 q^{8} + 4 q^{11} + 2 q^{13} - q^{16} + 2 q^{17} + 4 q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 768
$ \Gamma_0(N) $-optimal: no
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$(j)_{-}$
$3$ $2$ $I_{4}^{*}$ Additive -1 2 10 4
$5$ $2$ $I_{1}^{*}$ Additive 1 2 7 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 32.96.0.49

$p$-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 add add
$\lambda$-invariant(s) ? - -
$\mu$-invariant(s) ? - -

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

An entry ? indicates that the invariants have not yet been computed.

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 $d=$ 2, 4, 8 and 16.
Its isogeny class 225.b 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
$2$ \(\Q(\sqrt{5}) \) \(\Z/2\Z \times \Z/2\Z\) 2.2.5.1-405.1-a9
$2$ \(\Q(\sqrt{15}) \) \(\Z/4\Z\) 2.2.60.1-15.1-a9
$2$ \(\Q(\sqrt{3}) \) \(\Z/4\Z\) 2.2.12.1-1875.1-c8
$4$ \(\Q(\sqrt{3}, \sqrt{5})\) \(\Z/2\Z \times \Z/4\Z\) 4.4.3600.1-225.1-c2
$4$ \(\Q(\sqrt{6}, \sqrt{10})\) \(\Z/8\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{15})\) \(\Z/8\Z\) Not in database
$8$ 8.0.324000000.4 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.82944000000.32 \(\Z/8\Z\) Not in database
$8$ 8.8.3317760000.1 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.8.21233664000000.3 \(\Z/16\Z\) Not in database
$8$ 8.8.21233664000000.2 \(\Z/16\Z\) Not in database
$8$ 8.2.2767921875.2 \(\Z/6\Z\) Not in database
$16$ 16.0.26873856000000000000.6 \(\Z/4\Z \times \Z/4\Z\) 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 \(\Z/16\Z\) Not in database
$16$ 16.16.450868486864896000000000000.1 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/32\Z\) 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.