# Properties

 Label 825a1 Conductor $825$ Discriminant $-81675$ j-invariant $$-\frac{56197120}{3267}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+y=x^3-x^2-23x+53$$ y^2+y=x^3-x^2-23x+53 (homogenize, simplify) $$y^2z+yz^2=x^3-x^2z-23xz^2+53z^3$$ y^2z+yz^2=x^3-x^2z-23xz^2+53z^3 (dehomogenize, simplify) $$y^2=x^3-30240x+2123280$$ y^2=x^3-30240x+2123280 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 1, -23, 53])

gp: E = ellinit([0, -1, 1, -23, 53])

magma: E := EllipticCurve([0, -1, 1, -23, 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);

 $P$ = $$\left(1, 5\right)$$ (1, 5) $\hat{h}(P)$ ≈ $0.26728146468235865198494989376$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1, 5\right)$$, $$\left(1, -6\right)$$, $$\left(3, 1\right)$$, $$\left(3, -2\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$825$$ = $3 \cdot 5^{2} \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-81675$ = $-1 \cdot 3^{3} \cdot 5^{2} \cdot 11^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{56197120}{3267}$$ = $-1 \cdot 2^{15} \cdot 3^{-3} \cdot 5 \cdot 7^{3} \cdot 11^{-2}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.30121080929250401904572205629\dots$ Stable Faltings height: $-0.56945046136485408147918194516\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.26728146468235865198494989376\dots$ 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: $3.3747765339753457527067305386\dots$ 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: $2$  = $1\cdot1\cdot2$ 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) 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: $L'(E,1)$ ≈ $1.8040304299531682383269905694$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 72 $\Gamma_0(N)$-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$(j)_{-}$
$3$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$5$ $1$ $II$ Additive 1 2 2 0
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 3B 3.4.0.1
sage: gens = [[6, 1, 13, 24], [1, 0, 6, 1], [4, 3, 9, 7], [25, 6, 24, 7], [3, 4, 8, 11], [1, 6, 0, 1], [21, 2, 10, 7]]

sage: GL(2,Integers(30)).subgroup(gens)

magma: Gens := [[6, 1, 13, 24], [1, 0, 6, 1], [4, 3, 9, 7], [25, 6, 24, 7], [3, 4, 8, 11], [1, 6, 0, 1], [21, 2, 10, 7]];

magma: sub<GL(2,Integers(30))|Gens>;

The image of the adelic Galois representation has level $30$, index $16$, genus $0$, and generators

$\left(\begin{array}{rr} 6 & 1 \\ 13 & 24 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 25 & 6 \\ 24 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 2 \\ 10 & 7 \end{array}\right)$

## $p$-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 ss nonsplit add ord nonsplit ord ord ord ord ord ord ord ord ord ss 1,2 1 - 1 1 1 1 1 1 1 1 1 1 1 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 825a consists of 2 curves linked by isogenies of degree 3.

## 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 $K$ $2$ $$\Q(\sqrt{5})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.300.1 $$\Z/2\Z$$ Not in database $6$ 6.0.270000.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ 6.0.1235334375.1 $$\Z/3\Z$$ Not in database $6$ 6.2.450000.1 $$\Z/6\Z$$ Not in database $12$ deg 12 $$\Z/4\Z$$ Not in database $12$ deg 12 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $12$ 12.0.1822500000000.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $18$ 18.6.1346880880237797416008575439453125.3 $$\Z/9\Z$$ Not in database $18$ 18.0.7721710717374930375000000000000.2 $$\Z/6\Z$$ 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.