# Properties

 Label 7098.bc1 Conductor $7098$ Discriminant $113433535488$ j-invariant $$\frac{124318741396429}{51631104}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3-13517x+603537$$ y^2+xy=x^3-13517x+603537 (homogenize, simplify) $$y^2z+xyz=x^3-13517xz^2+603537z^3$$ y^2z+xyz=x^3-13517xz^2+603537z^3 (dehomogenize, simplify) $$y^2=x^3-17518059x+28211176422$$ y^2=x^3-17518059x+28211176422 (homogenize, minimize)

sage: E = EllipticCurve([1, 0, 0, -13517, 603537])

gp: E = ellinit([1, 0, 0, -13517, 603537])

magma: E := EllipticCurve([1, 0, 0, -13517, 603537]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z/{2}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(66, -33\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(66, -33\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$7098$$ = $2 \cdot 3 \cdot 7 \cdot 13^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $113433535488$ = $2^{10} \cdot 3 \cdot 7^{5} \cdot 13^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{124318741396429}{51631104}$$ = $2^{-10} \cdot 3^{-1} \cdot 7^{-5} \cdot 29^{3} \cdot 1721^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.0830459330521300522587451908\dots$ Stable Faltings height: $0.44180859368674586824537333041\dots$

## 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: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $1.0354952541232427161523397505\dots$ 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: $20$  = $( 2 \cdot 5 )\cdot1\cdot1\cdot2$ 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: $1$ (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: $L(E,1)$ ≈ $5.1774762706162135807616987525$

## 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);

$$q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + 2 q^{10} + q^{12} - q^{14} + 2 q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 12000 $\Gamma_0(N)$-optimal: no Manin constant: 1

## 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$(j)_{-}$
$2$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$7$ $1$ $I_{5}$ Non-split multiplicative 1 1 5 5
$13$ $2$ $III$ Additive -1 2 3 0

## 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
$2$ 2B 2.3.0.1
$5$ 5B 5.6.0.1
sage: gens = [[10901, 20, 10900, 21], [4696, 5, 9315, 10906], [8737, 20, 0, 1], [8191, 20, 0, 1], [5461, 20, 10, 201], [1, 10, 10, 101], [3656, 5, 10875, 10906], [1, 0, 20, 1], [11, 16, 10680, 10571], [10098, 5, 5335, 82], [1, 20, 0, 1]]

sage: GL(2,Integers(10920)).subgroup(gens)

magma: Gens := [[10901, 20, 10900, 21], [4696, 5, 9315, 10906], [8737, 20, 0, 1], [8191, 20, 0, 1], [5461, 20, 10, 201], [1, 10, 10, 101], [3656, 5, 10875, 10906], [1, 0, 20, 1], [11, 16, 10680, 10571], [10098, 5, 5335, 82], [1, 20, 0, 1]];

magma: sub<GL(2,Integers(10920))|Gens>;

The image of the adelic Galois representation has level $10920$, index $288$, genus $5$, and generators

$\left(\begin{array}{rr} 10901 & 20 \\ 10900 & 21 \end{array}\right),\left(\begin{array}{rr} 4696 & 5 \\ 9315 & 10906 \end{array}\right),\left(\begin{array}{rr} 8737 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8191 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5461 & 20 \\ 10 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 3656 & 5 \\ 10875 & 10906 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 10680 & 10571 \end{array}\right),\left(\begin{array}{rr} 10098 & 5 \\ 5335 & 82 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$

## $p$-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 13 split split ord nonsplit add 1 3 2 0 - 0 0 0 0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

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, 5 and 10.
Its isogeny class 7098.bc consists of 4 curves linked by isogenies of degrees dividing 10.

## 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 $K$ $2$ $$\Q(\sqrt{273})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.2952768.3 $$\Z/4\Z$$ Not in database $4$ 4.4.274625.1 $$\Z/10\Z$$ Not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.3845007938064384.123 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ deg 8 $$\Z/6\Z$$ Not in database $8$ 8.8.14667541267640625.3 $$\Z/2\Z \oplus \Z/10\Z$$ Not in database $16$ deg 16 $$\Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/20\Z$$ Not in database $20$ 20.0.210131154605237377668170642852783203125.1 $$\Z/10\Z$$ 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.