Properties

Label 303450.bb4
Conductor $303450$
Discriminant $1.305\times 10^{20}$
j-invariant \( \frac{5602762882081}{345888060} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -2673400, 1589072500]) # or
 
sage: E = EllipticCurve("303450.bb4")
 
gp: E = ellinit([1, 1, 0, -2673400, 1589072500]) \\ or
 
gp: E = ellinit("303450.bb4")
 
magma: E := EllipticCurve([1, 1, 0, -2673400, 1589072500]); // or
 
magma: E := EllipticCurve("303450.bb4");
 

\(y^2+xy=x^3+x^2-2673400x+1589072500\)

Mordell-Weil group structure

\(\Z^2 \times \Z/{2}\Z\)

Infinite order Mordell-Weil generators and heights

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \( \left(690, 8230\right) \)\( \left(375, 25100\right) \)
\(\hat{h}(P)\) ≈  $0.94199543587984547000970232464$$1.0386556880389942168272509359$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(\frac{4475}{4}, -\frac{4475}{8}\right) \)

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-1410, 51280\right) \), \( \left(-1410, -49870\right) \), \( \left(-360, 50230\right) \), \( \left(-360, -49870\right) \), \( \left(-339, 49733\right) \), \( \left(-339, -49394\right) \), \( \left(375, 25100\right) \), \( \left(375, -25475\right) \), \( \left(690, 8230\right) \), \( \left(690, -8920\right) \), \( \left(1229, 12101\right) \), \( \left(1229, -13330\right) \), \( \left(1769, 48101\right) \), \( \left(1769, -49870\right) \), \( \left(2840, 128630\right) \), \( \left(2840, -131470\right) \), \( \left(3875, 220400\right) \), \( \left(3875, -224275\right) \), \( \left(10490, 1056830\right) \), \( \left(10490, -1067320\right) \), \( \left(22501490, 106726224680\right) \), \( \left(22501490, -106748726170\right) \)

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: \(130451514289470937500 \)  =  \(2^{2} \cdot 3 \cdot 5^{7} \cdot 7^{8} \cdot 17^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{5602762882081}{345888060} \)  =  \(2^{-2} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-8} \cdot 17761^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.81246146368941272655476045431\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.18193766978892450286635714512\)
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: \( 256 \)  = \( 2\cdot1\cdot2^{2}\cdot2^{3}\cdot2^{2} \)
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 303450.2.a.bb

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 15728640
\( \Gamma_0(N) \)-optimal: no
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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 9.4603101118048414584032211254244581603 \)

Local data

This elliptic curve is 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\((j)_{-}\)
\(2\) \(2\) \(I_{2}\) Non-split multiplicative 1 1 2 2
\(3\) \(1\) \(I_{1}\) Non-split multiplicative 1 1 1 1
\(5\) \(4\) \(I_1^{*}\) Additive 1 2 7 1
\(7\) \(8\) \(I_{8}\) Split multiplicative -1 1 8 8
\(17\) \(4\) \(I_0^{*}\) 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 X85.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 3 \end{array}\right)$ and has index 24.

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(3,20) 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 non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 303450.bb consists of 4 curves linked by isogenies of degrees dividing 8.

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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{15}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{85}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{51}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{15}, \sqrt{51})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{30}, \sqrt{85})\) \(\Z/8\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{85})\) \(\Z/8\Z\) Not in database
$8$ 8.0.243547236000000.39 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ 8.8.277102632960000.4 \(\Z/2\Z \times \Z/8\Z\) 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 \(\Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/16\Z\) 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.