# Properties

 Label 912.b4 Conductor $912$ Discriminant $-1601384448$ j-invariant $$\frac{67419143}{390963}$$ CM no Rank $1$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x^2+136x-1872$$ y^2=x^3-x^2+136x-1872 (homogenize, simplify) $$y^2z=x^3-x^2z+136xz^2-1872z^3$$ y^2z=x^3-x^2z+136xz^2-1872z^3 (dehomogenize, simplify) $$y^2=x^3+10989x-1331694$$ y^2=x^3+10989x-1331694 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, 136, -1872])

gp: E = ellinit([0, -1, 0, 136, -1872])

magma: E := EllipticCurve([0, -1, 0, 136, -1872]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z \oplus \Z/{4}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(18, 78\right)$$ (18, 78) $\hat{h}(P)$ ≈ $2.6018939134308192452519184243$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(28, 152\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(9, 0\right)$$, $$(18,\pm 78)$$, $$(28,\pm 152)$$, $$(484,\pm 10640)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$912$$ = $2^{4} \cdot 3 \cdot 19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-1601384448$ = $-1 \cdot 2^{12} \cdot 3 \cdot 19^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{67419143}{390963}$$ = $3^{-1} \cdot 11^{3} \cdot 19^{-4} \cdot 37^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.44813851835711921574779975159\dots$ Stable Faltings height: $-0.24500866220282609366943236987\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $2.6018939134308192452519184243\dots$ 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: $0.75202298051859802360121969732\dots$ 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: $16$  = $2^{2}\cdot1\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ 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: $L'(E,1)$ ≈ $1.9566840157714437537809644231$

## 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);

$$q - q^{3} - 2 q^{5} + q^{9} + 6 q^{13} + 2 q^{15} - 6 q^{17} + q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 384 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$2$ $4$ $I_{4}^{*}$ Additive -1 4 12 0
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$19$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

## 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 4.12.0.7
sage: gens = [[277, 284, 238, 51], [7, 6, 450, 451], [97, 8, 388, 33], [1, 0, 8, 1], [449, 8, 448, 9], [403, 402, 298, 67], [160, 3, 157, 2], [1, 8, 0, 1], [1, 4, 4, 17]]

sage: GL(2,Integers(456)).subgroup(gens)

magma: Gens := [[277, 284, 238, 51], [7, 6, 450, 451], [97, 8, 388, 33], [1, 0, 8, 1], [449, 8, 448, 9], [403, 402, 298, 67], [160, 3, 157, 2], [1, 8, 0, 1], [1, 4, 4, 17]];

magma: sub<GL(2,Integers(456))|Gens>;

The image of the adelic Galois representation has level $456$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 277 & 284 \\ 238 & 51 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 450 & 451 \end{array}\right),\left(\begin{array}{rr} 97 & 8 \\ 388 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 449 & 8 \\ 448 & 9 \end{array}\right),\left(\begin{array}{rr} 403 & 402 \\ 298 & 67 \end{array}\right),\left(\begin{array}{rr} 160 & 3 \\ 157 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

## 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 ord ss ss ord ord split ord ord ord ord ord ord ord - 1 3 1,5 1,1 3 1 2 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 $d=$ 2 and 4.
Its isogeny class 912.b 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ 4.2.277248.1 $$\Z/8\Z$$ Not in database $8$ 8.0.2985984.1 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.389136420864.3 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.691798081536.10 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.5910009391872.5 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ 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.