# Properties

 Label 3025.d2 Conductor $3025$ Discriminant $-20796875$ j-invariant $$-32768$$ CM yes ($$D=-11$$) Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -183, 919])

gp: E = ellinit([0, 1, 1, -183, 919])

magma: E := EllipticCurve([0, 1, 1, -183, 919]);

$$y^2+y=x^3+x^2-183x+919$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(7, 5\right)$$ $$\hat{h}(P)$$ ≈ $0.75377822559486338339955515665$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-11, 41\right)$$, $$\left(-11, -42\right)$$, $$\left(7, 5\right)$$, $$\left(7, -6\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$3025$$ = $$5^{2} \cdot 11^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-20796875$$ = $$-1 \cdot 5^{6} \cdot 11^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-32768$$ = $$-1 \cdot 2^{15}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-11})/2]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $$0.18164613188757573762920830647\dots$$ Stable Faltings height: $$-1.2225466425290670856866572546\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.75377822559486338339955515665\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$2.1477081065172769177445555858\dots$$ 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: $$2$$  = $$1\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ 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[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q + q^{3} - 2q^{4} - 2q^{9} - 2q^{12} + 4q^{16} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 432 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

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

$$L'(E,1)$$ ≈ $$3.2377912112525936753769633141331257185$$

## 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$$(j)_{-}$$
$$5$$ $$1$$ $$I_0^{*}$$ Additive 1 2 6 0
$$11$$ $$2$$ $$III$$ Additive 1 2 3 0

## 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 mod $$p$$ Galois representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois representation
$$11$$ B.10.3

For all other primes $$p$$, the image is the normalizer of a split Cartan subgroup if $$\left(\frac{ -11 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -11 }{p}\right)=-1$$.

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge 5$$ of good ordinary reduction.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ss ordinary add ss add ss ss ss ordinary ss ordinary ordinary ss ss ordinary ? 3 - 1,1 - 1,1 1,1 1,1 1 1,1 3 1 1,1 1,1 1 ? 0 - 0,0 - 0,0 0,0 0,0 0 0,0 0 0 0,0 0,0 0

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 $$d=$$ 11.
Its isogeny class 3025.d consists of 2 curves linked by isogenies of degree 11.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.44.1 $$\Z/2\Z$$ Not in database $4$ 4.0.99825.1 $$\Z/3\Z$$ Not in database $4$ 4.2.299475.1 $$\Z/3\Z$$ Not in database $6$ 6.0.21296.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.0.89685275625.4 $$\Z/3\Z \times \Z/3\Z$$ Not in database $8$ 8.0.5536128125.1 $$\Z/5\Z$$ Not in database $10$ 10.10.669871503125.1 $$\Z/11\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/6\Z$$ Not in database $16$ 16.4.766217865410400390625.1 $$\Z/5\Z$$ Not in database $16$ Deg 16 $$\Z/15\Z$$ 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.