# Properties

 Label 59290br1 Conductor $59290$ Discriminant $-5.836\times 10^{14}$ j-invariant $$\frac{1367631}{2800}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, 13711, -987827])

gp: E = ellinit([1, -1, 0, 13711, -987827])

magma: E := EllipticCurve([1, -1, 0, 13711, -987827]);

$$y^2+xy=x^3-x^2+13711x-987827$$

## Mordell-Weil group structure

$$\Z^2 \times \Z/{2}\Z$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(102, 1159\right)$$ $$\left(234, 3755\right)$$ $$\hat{h}(P)$$ ≈ $1.3510880653707850192590807483$ $2.6466365610342602446538555382$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(58, -29\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(58, -29\right)$$, $$\left(102, 1159\right)$$, $$\left(102, -1261\right)$$, $$\left(139, 1825\right)$$, $$\left(139, -1964\right)$$, $$\left(234, 3755\right)$$, $$\left(234, -3989\right)$$, $$\left(597, 14524\right)$$, $$\left(597, -15121\right)$$, $$\left(1147, 38449\right)$$, $$\left(1147, -39596\right)$$, $$\left(1822, 76999\right)$$, $$\left(1822, -78821\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$59290$$ = $$2 \cdot 5 \cdot 7^{2} \cdot 11^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-583582664249200$$ = $$-1 \cdot 2^{4} \cdot 5^{2} \cdot 7^{7} \cdot 11^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1367631}{2800}$$ = $$2^{-4} \cdot 3^{3} \cdot 5^{-2} \cdot 7^{-1} \cdot 37^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.5177432019815328615641220235\dots$$ Stable Faltings height: $$-0.65415950894530906301952613720\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$3.5733158879574577634230828203\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.26901632746305495906962503217\dots$$ 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: $$32$$  = $$2\cdot2\cdot2\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

Modular form 59290.2.a.bc

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);

$$q - q^{2} + q^{4} + q^{5} - q^{8} - 3q^{9} - q^{10} - 6q^{13} + q^{16} + 2q^{17} + 3q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 245760 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$7.6902425363496036961175529549787794263$$

## Local data

This elliptic curve is not 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$7$$ $$2$$ $$I_1^{*}$$ Additive -1 2 7 1
$$11$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 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 X34.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 0 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 7 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 4 & 3 \end{array}\right)$ and has index 12.

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
$$2$$ B

## $p$-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 nonsplit ss split add add ordinary ordinary ss ss ordinary ordinary ordinary ordinary ordinary ordinary 5 2,2 3 - - 2 2 2,2 2,2 2 2 2 2 2 2 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 $$d=$$ 2 and 4.
Its isogeny class 59290br consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{77})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-11})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-7}, \sqrt{-11})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.275599841440000.16 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.4.89991784960000.37 $$\Z/8\Z$$ Not in database $8$ 8.0.7055355940864.55 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ 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.