Properties

Label 11310.b1
Conductor $11310$
Discriminant $31758480$
j-invariant \( \frac{13701674594089}{31758480} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -498, 4068])
 
gp: E = ellinit([1, 1, 0, -498, 4068])
 
magma: E := EllipticCurve([1, 1, 0, -498, 4068]);
 

\(y^2+xy=x^3+x^2-498x+4068\)  Toggle raw display

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

$P$ =  \(\left(8, 22\right)\)  Toggle raw display\(\left(11, 3\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.45571574077937242924157035397$$2.0347894923805044242342517253$

Torsion generators

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

\( \left(12, -6\right) \)  Toggle raw display

Integral points

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

\( \left(-24, 66\right) \), \( \left(-24, -42\right) \), \( \left(-13, 99\right) \), \( \left(-13, -86\right) \), \( \left(8, 22\right) \), \( \left(8, -30\right) \), \( \left(11, 3\right) \), \( \left(11, -14\right) \), \( \left(12, -6\right) \), \( \left(13, -5\right) \), \( \left(13, -8\right) \), \( \left(21, 48\right) \), \( \left(21, -69\right) \), \( \left(48, 282\right) \), \( \left(48, -330\right) \), \( \left(112, 1114\right) \), \( \left(112, -1226\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 11310 \)  =  $2 \cdot 3 \cdot 5 \cdot 13 \cdot 29$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $31758480 $  =  $2^{4} \cdot 3^{4} \cdot 5 \cdot 13^{2} \cdot 29 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{13701674594089}{31758480} \)  =  $2^{-4} \cdot 3^{-4} \cdot 5^{-1} \cdot 13^{-2} \cdot 29^{-1} \cdot 23929^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.32019318621943712247829609961\dots$
Stable Faltings height: $0.32019318621943712247829609961\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $2$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.91274465897100494269767030360\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $2.0864333223694759558757064806\dots$
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: $ 8 $  = $ 2\cdot2\cdot1\cdot2\cdot1 $
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)
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^{(2)}(E,1)/2! $ ≈ $ 3.8087617425837362992297553494367986876 $

Modular invariants

Modular form 11310.2.a.b

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);
 

\( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9} + q^{10} - 2 q^{11} - q^{12} + q^{13} + 2 q^{14} + q^{15} + q^{16} - 6 q^{17} - q^{18} - 2 q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 5632
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

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$(j)_{-}$
$2$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$13$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$29$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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 4.6.0.3

$p$-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 nonsplit nonsplit nonsplit ordinary ordinary split ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 6 2 2 2 2 3 2 2 4 2 2 2 2 2 2
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 11310.b consists of 2 curves linked by isogenies of degree 2.

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
$2$ \(\Q(\sqrt{145}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$4$ 4.0.2320.2 \(\Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.113164960000.3 \(\Z/2\Z \times \Z/4\Z\) 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 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) 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.