# Properties

 Label 303450.cm1 Conductor $303450$ Discriminant $1.239\times 10^{12}$ j-invariant $$\frac{18694465225}{6858432}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -3126, -40952])

gp: E = ellinit([1, 0, 1, -3126, -40952])

magma: E := EllipticCurve([1, 0, 1, -3126, -40952]);

$$y^2+xy+y=x^3-3126x-40952$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-28, 171\right)$$ $$\left(77, 381\right)$$ $$\hat{h}(P)$$ ≈ $0.38945604712963082450153941191$ $0.63996125536368144888603179552$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-43, 141\right)$$, $$\left(-43, -99\right)$$, $$\left(-28, 171\right)$$, $$\left(-28, -144\right)$$, $$\left(-19, 117\right)$$, $$\left(-19, -99\right)$$, $$\left(-14, 17\right)$$, $$\left(-14, -4\right)$$, $$\left(62, 36\right)$$, $$\left(62, -99\right)$$, $$\left(77, 381\right)$$, $$\left(77, -459\right)$$, $$\left(128, 1224\right)$$, $$\left(128, -1353\right)$$, $$\left(152, 1656\right)$$, $$\left(152, -1809\right)$$, $$\left(413, 8109\right)$$, $$\left(413, -8523\right)$$, $$\left(917, 27261\right)$$, $$\left(917, -28179\right)$$, $$\left(3782, 230676\right)$$, $$\left(3782, -234459\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$303450$$ = $$2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$1238804280000$$ = $$2^{6} \cdot 3^{7} \cdot 5^{4} \cdot 7^{2} \cdot 17^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{18694465225}{6858432}$$ = $$2^{-6} \cdot 3^{-7} \cdot 5^{2} \cdot 7^{-2} \cdot 17 \cdot 353^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.0206793653723938209797525625\dots$$ Stable Faltings height: $$0.011997837218324349404577015113\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.24923676933640684824651547916\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.65816878296845467742193402665\dots$$ 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: $$84$$  = $$2\cdot7\cdot3\cdot2\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 303450.2.a.cm

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);

$$q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} - q^{8} + q^{9} - 4q^{11} + q^{12} + 3q^{13} - q^{14} + q^{16} - q^{18} - 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 798336 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$13.779348336191118123764395275420065772$$

## Local data

This elliptic curve is not semistable. There are 5 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$3$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$5$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$17$$ $$1$$ $$II$$ Additive 1 2 2 0

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 $$p$$ .

## $p$-adic data

### $p$-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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 303450.cm consists of this curve only.

## 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 $K$ $3$ 3.3.86700.1 $$\Z/2\Z$$ Not in database $6$ 6.6.90202680000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.