# Properties

 Label 1155n2 Conductor $1155$ Discriminant $-4.386\times 10^{17}$ j-invariant $$-\frac{2126464142970105856}{438611057788643355}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -26790, -31917424])

gp: E = ellinit([0, 1, 1, -26790, -31917424])

magma: E := EllipticCurve([0, 1, 1, -26790, -31917424]);

$$y^2+y=x^3+x^2-26790x-31917424$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{6197}{4}, \frac{483149}{8}\right)$$ $\hat{h}(P)$ ≈ $0.86625097826011141373894683439$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1155$$ = $3 \cdot 5 \cdot 7 \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-438611057788643355$ = $-1 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{15}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{2126464142970105856}{438611057788643355}$$ = $-1 \cdot 2^{12} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-15} \cdot 179^{3} \cdot 449^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.0644840318921348788053224715\dots$ Stable Faltings height: $2.0644840318921348788053224715\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.86625097826011141373894683439\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.13267092749352958309939093999\dots$ 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: $15$  = $1\cdot1\cdot1\cdot( 3 \cdot 5 )$ 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) 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: $L'(E,1)$ ≈ $1.7238948109191946881626844460878985447$

## 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);

$$q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} + q^{7} + q^{9} - 2q^{10} + q^{11} + 2q^{12} - 6q^{13} - 2q^{14} + q^{15} - 4q^{16} - 7q^{17} - 2q^{18} - 5q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 30000 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is semistable. There are 4 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$(j)_{-}$
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$5$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$7$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$11$ $15$ $I_{15}$ Split multiplicative -1 1 15 15

## 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
$5$ 5B.1.2 5.24.0.3

## $p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

## 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 split split split split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 8,1 4 6 2 2 1 1 1 1 1 1 1 1 1 1 0,0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 5.
Its isogeny class 1155n consists of 2 curves linked by isogenies of degree 5.

## 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 $K$ $3$ 3.1.4620.1 $$\Z/2\Z$$ Not in database $4$ $$\Q(\zeta_{5})$$ $$\Z/5\Z$$ Not in database $5$ 5.1.379845703125.1 $$\Z/5\Z$$ Not in database $6$ 6.0.24652782000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.265831216875.2 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/10\Z$$ Not in database $15$ 15.1.949017750063435680904808044433593750000000000.1 $$\Z/10\Z$$ Not in database $20$ 20.0.2602189288594646001585715566761791706085205078125.6 $$\Z/5\Z \times \Z/5\Z$$ 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.