Properties

Label 3025a1
Conductor $3025$
Discriminant $-20796875$
j-invariant \( -32768 \)
CM yes (\(D=-11\))
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -183, 919])
 
gp: E = ellinit([0, 1, 1, -183, 919])
 
magma: E := EllipticCurve([0, 1, 1, -183, 919]);
 

\(y^2+y=x^3+x^2-183x+919\)  Toggle raw display

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \(\left(7, 5\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $0.75377822559486338339955515665$

Integral points

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

\( \left(-11, 41\right) \), \( \left(-11, -42\right) \), \( \left(7, 5\right) \), \( \left(7, -6\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 3025 \)  =  \(5^{2} \cdot 11^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-20796875 \)  =  \(-1 \cdot 5^{6} \cdot 11^{3} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -32768 \)  =  \(-1 \cdot 2^{15}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-11})/2]\) (potential complex multiplication)
Sato-Tate group: $N(\mathrm{U}(1))$
Faltings height: \(0.18164613188757573762920830647\dots\)
Stable Faltings height: \(-1.2225466425290670856866572546\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.75377822559486338339955515665\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(2.1477081065172769177445555858\dots\)
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: \( 2 \)  = \( 1\cdot2 \)
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: \(1\) (exact)

Modular invariants

Modular form   3025.2.a.d

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);
 

\( q + q^{3} - 2q^{4} - 2q^{9} - 2q^{12} + 4q^{16} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 432
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 3.2377912112525936753769633141331257185 \)

Local data

This elliptic curve is not semistable. There are 2 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\((j)_{-}\)
\(5\) \(1\) \(I_0^{*}\) Additive 1 2 6 0
\(11\) \(2\) \(III\) 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 mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois representation
\(11\) B.10.3

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=-1\).

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss ordinary add ss add ss ss ss ordinary ss ordinary ordinary ss ss ordinary
$\lambda$-invariant(s) ? 3 - 1,1 - 1,1 1,1 1,1 1 1,1 3 1 1,1 1,1 1
$\mu$-invariant(s) ? 0 - 0,0 - 0,0 0,0 0,0 0 0,0 0 0 0,0 0,0 0

An entry ? indicates that the invariants have not yet been computed.

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 \(d=\) 11.
Its isogeny class 3025a consists of 2 curves linked by isogenies of degree 11.

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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.44.1 \(\Z/2\Z\) Not in database
$4$ 4.0.99825.1 \(\Z/3\Z\) Not in database
$4$ 4.2.299475.1 \(\Z/3\Z\) Not in database
$6$ 6.0.21296.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ 8.0.89685275625.4 \(\Z/3\Z \times \Z/3\Z\) Not in database
$8$ 8.0.5536128125.1 \(\Z/5\Z\) Not in database
$10$ 10.10.669871503125.1 \(\Z/11\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/9\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z\) Not in database
$16$ 16.4.766217865410400390625.1 \(\Z/5\Z\) Not in database
$16$ Deg 16 \(\Z/15\Z\) 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.