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This is a model for the modular curve $X_0(15)$.

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -10, -10])

gp: E = ellinit([1, 1, 1, -10, -10])

magma: E := EllipticCurve([1, 1, 1, -10, -10]);

$$y^2+xy+y=x^3+x^2-10x-10$$ ## Mordell-Weil group structure

$$\Z/{2}\Z \times \Z/{4}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-1, 0\right)$$, $$\left(8, 18\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2, 3\right)$$, $$\left(-2, -2\right)$$, $$\left(-1, 0\right)$$, $$\left(3, -2\right)$$, $$\left(8, 18\right)$$, $$\left(8, -27\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$15$$ = $$3 \cdot 5$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$50625$$ = $$3^{4} \cdot 5^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{111284641}{50625}$$ = $$3^{-4} \cdot 5^{-4} \cdot 13^{3} \cdot 37^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$-0.40227757208284378744359649716\dots$$ Stable Faltings height: $$-0.40227757208284378744359649716\dots$$

## 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  magma: RealPeriod(E); Real period: $$2.8012060846652040463603616736\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$8$$  = $$2\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$8$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{15} - q^{16} + 2q^{17} - q^{18} + 4q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L(E,1)$$ ≈ $$0.35015076058315050579504520920239987269$$

## Local data

This elliptic curve is semistable. There are 2 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

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

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X187d.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 4 \\ 2 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 5 \end{array}\right)$ and has index 96.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ Cs

## $p$-adic data

### $p$-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

$p$ Reduction type $\lambda$-invariant(s) 2 3 5 ordinary nonsplit split 0 0 1 1 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 and 4.
Its isogeny class 15a consists of 8 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z \times \Z/4\Z$$ 2.0.4.1-225.2-a6 $2$ $$\Q(\sqrt{5})$$ $$\Z/2\Z \times \Z/8\Z$$ 2.2.5.1-45.1-a5 $4$ $$\Q(i, \sqrt{5})$$ $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ $$\Q(\zeta_{24})$$ $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.0.3317760000.5 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.4.324000000.3 $$\Z/2\Z \times \Z/16\Z$$ Not in database $8$ 8.2.110716875.2 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ 16.0.11007531417600000000.1 $$\Z/8\Z \times \Z/8\Z$$ Not in database $16$ 16.0.26873856000000000000.10 $$\Z/4\Z \times \Z/16\Z$$ Not in database $16$ 16.0.1048576000000000000.6 $$\Z/4\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/24\Z$$ 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.