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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -155115, 23514138])

gp: E = ellinit([0, 0, 0, -155115, 23514138])

magma: E := EllipticCurve([0, 0, 0, -155115, 23514138]);

$$y^2=x^3-155115x+23514138$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$1296$$ = $$2^{4} \cdot 3^{4}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-278628139008$$ = $$-1 \cdot 2^{19} \cdot 3^{12}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{189613868625}{128}$$ = $$-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 383^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## 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: $$0.80848602428825605028754866728$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$2$$  = $$2\cdot1$$ sage: E.torsion_order()  gp: elltors(E)  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)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - 2q^{7} + 3q^{11} + 2q^{13} - 3q^{17} + q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3024 $$\Gamma_0(N)$$-optimal: no 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);

$$L(E,1)$$ ≈ $$1.6169720485765121005750973345518439593$$

## Local data

This elliptic curve is not 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{11}^{*}$$ Additive -1 4 19 7
$$3$$ $$1$$ $$II^{*}$$ Additive 1 4 12 0

## 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 X4.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.

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
$$3$$ B
$$7$$ B

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

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 $$d=$$ 3, 7 and 21.
Its isogeny class 1296.g consists of 4 curves linked by isogenies of degrees dividing 21.

## 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 $K$ $2$ $$\Q(\sqrt{3})$$ $$\Z/3\Z$$ 2.2.12.1-1458.1-i1 $3$ 3.1.648.1 $$\Z/2\Z$$ Not in database $6$ 6.0.3359232.4 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.186624.1 $$\Z/3\Z$$ Not in database $6$ 6.6.7057326528.2 $$\Z/7\Z$$ Not in database $6$ 6.2.20155392.5 $$\Z/6\Z$$ Not in database $12$ 12.2.5777633090469888.3 $$\Z/4\Z$$ Not in database $12$ 12.0.313456656384.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ 12.0.6499837226778624.48 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ 12.12.448252719505312813056.1 $$\Z/21\Z$$ Not in database $18$ 18.6.1376809511370776442839236608.2 $$\Z/9\Z$$ Not in database $18$ 18.0.3537182715531733726396416.1 $$\Z/6\Z$$ Not in database $18$ 18.6.1439728439119866152968380211003392.1 $$\Z/14\Z$$ Not in database

We only show fields where the torsion growth is primitive.