Properties

 Label 97104.bc4 Conductor $97104$ Discriminant $-4.337\times 10^{17}$ j-invariant $$\frac{103823}{4386816}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, 4528, -31686720])

gp: E = ellinit([0, -1, 0, 4528, -31686720])

magma: E := EllipticCurve([0, -1, 0, 4528, -31686720]);

$$y^2=x^3-x^2+4528x-31686720$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(824, 23040\right)$$ $$\left(3674, 222630\right)$$ $$\hat{h}(P)$$ ≈ $2.9289919280826916409263120245$ $7.0192002632698589068967889112$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(312, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(312, 0\right)$$, $$(824,\pm 23040)$$, $$(890,\pm 26010)$$, $$(3674,\pm 222630)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$97104$$ = $$2^{4} \cdot 3 \cdot 7 \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-433713454654685184$$ = $$-1 \cdot 2^{24} \cdot 3^{2} \cdot 7 \cdot 17^{7}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{103823}{4386816}$$ = $$2^{-12} \cdot 3^{-2} \cdot 7^{-1} \cdot 17^{-1} \cdot 47^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.0631565425007701036518394697\dots$$ Stable Faltings height: $$-0.046597310087283245890159960695\dots$$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$19.408647279173549437489308617\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.13716083068503871660779786858\dots$$ 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: $$16$$  = $$2^{2}\cdot2\cdot1\cdot2$$ 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: $$1$$ (rounded)

Modular invariants

Modular form 97104.2.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);

$$q - q^{3} + 2q^{5} - q^{7} + q^{9} - 6q^{13} - 2q^{15} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1327104 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$10.648424733137442312442171928257071758$$

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$$(j)_{-}$$
$$2$$ $$4$$ $$I_{16}^{*}$$ Additive -1 4 24 12
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$17$$ $$2$$ $$I_1^{*}$$ Additive 1 2 7 1

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 add nonsplit ordinary nonsplit ss ordinary add ss ordinary ordinary ordinary ordinary ordinary ordinary ss - 2 2 2 2,2 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 97104.bc 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 $K$ $2$ $$\Q(\sqrt{-119})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{7})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-17})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{7}, \sqrt{-17})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.3680330068444176.11 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ 8.0.21724846424064.27 $$\Z/8\Z$$ 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.