# Properties

 Label 5265.m1 Conductor $5265$ Discriminant $-47385$ j-invariant $$-\frac{2146689}{65}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3-x^2-24x+53$$ y^2+xy=x^3-x^2-24x+53 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-24xz^2+53z^3$$ y^2z+xyz=x^3-x^2z-24xz^2+53z^3 (dehomogenize, simplify) $$y^2=x^3-387x+3006$$ y^2=x^3-387x+3006 (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 0, -24, 53])

gp: E = ellinit([1, -1, 0, -24, 53])

magma: E := EllipticCurve([1, -1, 0, -24, 53]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(4, 1\right)$$ (4, 1) $\hat{h}(P)$ ≈ $0.47801206173053277709212084978$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(4, 1\right)$$, $$\left(4, -5\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$5265$$ = $3^{4} \cdot 5 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-47385$ = $-1 \cdot 3^{6} \cdot 5 \cdot 13$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{2146689}{65}$$ = $-1 \cdot 3^{3} \cdot 5^{-1} \cdot 13^{-1} \cdot 43^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.32510314325975029050875268521\dots$ Stable Faltings height: $-0.87440928759380513620637530367\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.47801206173053277709212084978\dots$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $3.5662639899425718922397845316\dots$ 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: $3$  = $3\cdot1\cdot1$ 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: $L'(E,1)$ ≈ $5.1141516075234143947997510832$

## 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);

$$q + q^{2} - q^{4} + q^{5} + 2 q^{7} - 3 q^{8} + q^{10} + q^{11} - q^{13} + 2 q^{14} - q^{16} - 2 q^{17} - 3 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 360 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$3$ $3$ $IV$ Additive -1 4 6 0
$5$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 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 $\ell$.

sage: gens = [[157, 2, 157, 3], [1, 0, 2, 1], [1, 2, 0, 1], [41, 2, 41, 3], [131, 2, 131, 3], [259, 2, 258, 3], [1, 1, 259, 0]]

sage: GL(2,Integers(260)).subgroup(gens)

magma: Gens := [[157, 2, 157, 3], [1, 0, 2, 1], [1, 2, 0, 1], [41, 2, 41, 3], [131, 2, 131, 3], [259, 2, 258, 3], [1, 1, 259, 0]];

magma: sub<GL(2,Integers(260))|Gens>;

The image of the adelic Galois representation has level $260$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 157 & 2 \\ 157 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right),\left(\begin{array}{rr} 131 & 2 \\ 131 & 3 \end{array}\right),\left(\begin{array}{rr} 259 & 2 \\ 258 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 259 & 0 \end{array}\right)$

## $p$-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 ord add split ord ord nonsplit ord ord ss ord ord ord ss ord ord 24 - 2 3 1 1 1 3 1,1 1 1 1 1,3 1 1 0 - 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 no rational isogenies. Its isogeny class 5265.m consists of this curve only.

## 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 $K$ $3$ 3.1.21060.1 $$\Z/2\Z$$ Not in database $6$ 6.0.115316136000.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.39039316875.2 $$\Z/3\Z$$ Not in database $12$ deg 12 $$\Z/4\Z$$ 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.